Search: keyword:new
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1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 1, 4, 2, 1, 3, 1, 2, 1, 4, 2, 4, 2, 1, 1, 4, 4, 1, 2, 16, 3, 2, 3, 4, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 4, 8, 2, 1, 1, 6, 1, 9, 4, 2, 4, 2, 1, 2, 2, 1, 16, 4, 1, 1, 2, 2, 3, 4, 4, 12, 1, 1, 1, 2, 2, 4, 2, 8, 1, 11, 1, 2, 4, 3, 2, 2, 2, 3, 1, 4, 4, 16, 8, 2, 6, 7, 1, 2, 1, 1, 6, 4, 1, 4
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OFFSET
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1,3
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LINKS
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Table of n, a(n) for n=1..105.
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = A353783(n) / A080398(n).
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PROG
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(PARI)
A080398(n) = factorback(factor(sigma(n))[, 1]);
A353783(n) = { my(f=factor(n)~); lcm(vector(#f, i, sigma(f[1, i]^f[2, i]))); };
A353785(n) = (A353783(n) / A080398(n));
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CROSSREFS
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Cf. A000203, A080398, A353783.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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A353784
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a(n) = sigma(n) / LCM_{p^e||n} sigma(p^e), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.
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+0
0
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1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 4, 3, 2, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1, 3, 1, 1, 1, 1, 2, 7, 1, 1, 6, 1, 4, 3, 1, 2, 1, 1, 1, 1, 2, 12, 1, 1, 4, 6, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 3, 1, 4, 6, 1, 2, 3, 1, 3, 2, 1, 4, 3, 2, 1, 1, 3, 1, 1, 1, 6, 1, 1, 8
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OFFSET
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1,10
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LINKS
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Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = A000203(n) / A353783(n).
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MATHEMATICA
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Array[DivisorSigma[1, #]/(LCM @@ DivisorSigma[1, Power @@@ FactorInteger[#]]) &, 105] (* Michael De Vlieger, May 08 2022 *)
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PROG
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(PARI) A353784(n) = { my(f=factor(n)~); (sigma(n) / lcm(vector(#f, i, sigma(f[1, i]^f[2, i])))); };
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CROSSREFS
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Cf. A000203, A353783.
Cf. A336547 (positions of 1's), A336548 (of terms > 1).
Cf. also A345045, A345047
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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A353783
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a(n) = LCM_{p^e||n} sigma(p^e), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.
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+0
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1, 3, 4, 7, 6, 12, 8, 15, 13, 6, 12, 28, 14, 24, 12, 31, 18, 39, 20, 42, 8, 12, 24, 60, 31, 42, 40, 56, 30, 12, 32, 63, 12, 18, 24, 91, 38, 60, 28, 30, 42, 24, 44, 84, 78, 24, 48, 124, 57, 93, 36, 14, 54, 120, 12, 120, 20, 30, 60, 84, 62, 96, 104, 127, 42, 12, 68, 126, 24, 24, 72, 195, 74, 114, 124, 140, 24, 84, 80
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OFFSET
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1,2
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LINKS
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Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = A000203(n) / A353784(n).
a(n) = A353785(n) * A080398(n).
For all n >= 1, A087207(a(n)) = A351560(n).
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MATHEMATICA
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Array[LCM @@ DivisorSigma[1, Power @@@ FactorInteger[#]] &, 79] (* Michael De Vlieger, May 08 2022 *)
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PROG
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(PARI) A353783(n) = { my(f=factor(n)~); lcm(vector(#f, i, sigma(f[1, i]^f[2, i]))); };
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CROSSREFS
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Cf. A000203, A080398, A087207, A351560, A353784, A353785.
Cf. also A345044, A345046.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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A353807
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Numbers k such that A353802(k) / sigma(sigma(k)) is an integer > 1, where A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).
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+0
0
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1819, 5088, 7215, 7276, 9487, 9523, 11895, 13303, 14235, 16371, 20179, 21079, 21255, 24531, 24751, 24931, 25824, 29104, 30615, 32224, 33855, 36199, 37635, 37948, 38092, 38664, 40443, 40515, 41847, 43831, 44655, 45475, 45695, 45883, 46995, 48043, 48399, 53835, 54015, 54568, 55747, 56899, 56928, 59599, 60495, 61035
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OFFSET
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1,1
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COMMENTS
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Numbers k such that A353805(k) = 1, but A353806(k) > 1.
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LINKS
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Table of n, a(n) for n=1..46.
Index entries for sequences related to sigma(n)
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EXAMPLE
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A353802(1819) = 10920 = 2*A051027(1819) = 2*5460, therefore 1819 is included as a term.
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PROG
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(PARI)
A051027(n) = sigma(sigma(n));
A353805(n) = { my(f = factor(n)); (A051027(n) / gcd(A051027(n), prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])))); };
A353806(n) = { my(f = factor(n), u=prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); (u / gcd(A051027(n), u)); };
isA353807(n) = ((1==A353805(n)) && (1!=A353806(n)));
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CROSSREFS
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Cf. A000203, A051027, A353802, A353805, A353806.
Cf. also A336561.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 5, 16, 1, 1, 1, 1, 1, 1, 1, 112, 1, 1, 49, 13, 45, 1, 1, 1, 7, 16, 1, 5, 1, 1, 1, 16, 1, 1, 1, 1, 7, 64, 1, 1, 112, 1, 49, 16, 1, 7, 1, 1, 1, 1, 9, 784, 1, 1, 5, 720, 1, 1, 1, 1, 1, 1, 5, 7, 1, 1, 1, 16, 1, 5, 117, 1, 7, 16, 1, 16, 45, 1, 147, 16, 7
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OFFSET
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1,10
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COMMENTS
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Numerator of fraction A353802(n) / A051027(n).
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LINKS
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Table of n, a(n) for n=1..95.
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = A353802(n) / A353804(n) = A353802(n) / gcd(A051027(n), A353802(n)).
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PROG
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(PARI)
A051027(n) = sigma(sigma(n));
A353806(n) = { my(f = factor(n), u=prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); (u / gcd(A051027(n), u)); };
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CROSSREFS
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Cf. A000203, A051027, A353802, A353803, A353804, A353805 (denominators).
Cf. A336547 (positions of 1's), A336548 (positions of terms > 1), see also A353807.
Cf. also A353755, A353756.
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KEYWORD
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nonn,frac,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 31, 10, 31, 1, 1, 1, 5, 13, 1, 3, 1, 1, 1, 13, 1, 1, 1, 1, 5, 57, 1, 1, 65, 1, 31, 13, 1, 5, 1, 1, 1, 1, 7, 403, 1, 1, 3, 403, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 13, 1, 3, 70, 1, 5, 13, 1, 13, 31, 1, 85, 13, 5, 1, 1, 13
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OFFSET
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1,10
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COMMENTS
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Denominator of fraction A353802(n) / A051027(n).
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LINKS
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Table of n, a(n) for n=1..98.
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = A051027(n) / A353804(n).
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PROG
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(PARI)
A051027(n) = sigma(sigma(n));
A353805(n) = { my(f = factor(n)); (A051027(n) / gcd(A051027(n), prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])))); };
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CROSSREFS
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Cf. A000203, A051027, A353802, A353803, A353804, A353806 (numerators).
Positions of 1's is given by the union of A336547 and A353807.
Cf. also A353755, A353756.
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KEYWORD
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nonn,frac,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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A353804
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Greatest common divisor of sigma(sigma(n)) and Product_{p^e||n} sigma(sigma(p^e)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.
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+0
0
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1, 4, 7, 8, 12, 28, 15, 24, 14, 3, 28, 56, 24, 60, 12, 32, 39, 56, 42, 96, 21, 7, 60, 168, 32, 96, 90, 120, 72, 3, 63, 104, 4, 12, 4, 112, 60, 168, 24, 18, 96, 84, 84, 224, 168, 15, 124, 224, 80, 128, 39, 3, 120, 360, 3, 360, 6, 18, 168, 96, 96, 252, 210, 128, 32, 1, 126, 312, 84, 1, 195, 336, 114, 240, 224, 336
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OFFSET
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1,2
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LINKS
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Table of n, a(n) for n=1..76.
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = gcd(A051027(n), A353802(n)) = gcd(A051027(n), A353803(n)) = gcd(A353802(n), A353803(n)).
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PROG
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(PARI)
A051027(n) = sigma(sigma(n));
A353804(n) = { my(f = factor(n)); gcd(A051027(n), prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); };
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CROSSREFS
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Cf. A000203, A051027, A353802, A353803, A353805, A353806.
Cf. also A353754.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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A353803
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a(n) = Product_{p^e||n} sigma(sigma(p^e)) - sigma(sigma(n)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.
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+0
0
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0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 42, 21, 0, 0, 0, 0, 0, 0, 0, 141, 0, 0, 72, 36, 56, 0, 0, 0, 48, 54, 0, 168, 0, 0, 0, 45, 0, 0, 0, 0, 78, 21, 0, 0, 141, 0, 108, 54, 0, 192, 0, 0, 0, 0, 64, 381, 0, 0, 168, 317, 0, 0, 0, 0, 0, 0, 168, 192, 0, 0, 0, 72, 0, 336, 188, 0, 144, 126, 0, 126, 112
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OFFSET
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1,10
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LINKS
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Table of n, a(n) for n=1..91.
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = A353802(n) - A051027(n).
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PROG
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(PARI)
A051027(n) = sigma(sigma(n));
A353803(n) = { my(f = factor(n)); (prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])) - A051027(n)); };
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CROSSREFS
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Cf. A000203, A051027, A353802, A353804, A353805, A353806.
Cf. A336547 (positions of 0's), A336548 (positions of terms > 0).
Cf. also A353753.
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KEYWORD
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nonn,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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A353802
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Multiplicative with a(p^e) = sigma(sigma(p^e)).
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+0
0
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1, 4, 7, 8, 12, 28, 15, 24, 14, 48, 28, 56, 24, 60, 84, 32, 39, 56, 42, 96, 105, 112, 60, 168, 32, 96, 90, 120, 72, 336, 63, 104, 196, 156, 180, 112, 60, 168, 168, 288, 96, 420, 84, 224, 168, 240, 124, 224, 80, 128, 273, 192, 120, 360, 336, 360, 294, 288, 168, 672, 96, 252, 210, 128, 288, 784, 126, 312, 420, 720
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OFFSET
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1,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)
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FORMULA
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a(n) = Product_{p^e||n} sigma(sigma(p^e)), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.
a(n) = A353802(n) + A051027(n).
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PROG
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(PARI)
A051027(n) = sigma(sigma(n));
A353802(n) = { my(f = factor(n)); prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])); };
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CROSSREFS
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Cf. A000203, A051027, A353803, A353804, A353805, A353806.
Cf. also A353752.
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KEYWORD
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nonn,mult,new
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AUTHOR
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Antti Karttunen, May 08 2022
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STATUS
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approved
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A353620
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Side b of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.
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+0
0
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10, 48, 132, 195, 280, 357, 504, 510, 665, 792, 840, 840, 1035, 1288, 1485, 1575, 1740, 1848, 1872, 1890, 2233, 2496, 2604, 2610, 2640, 3003, 3069, 3520, 3536, 3885, 4095, 4368, 4560, 4620, 4662, 4680, 5291, 5712, 5904, 5928, 6006, 6579, 6765, 6992, 7462, 7480, 7568, 8037, 8385, 8415, 8820
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OFFSET
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1,1
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COMMENTS
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The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c.
In the case B = 3*C, the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
Equivalently, length of side opposite to the angle that is the triple of an other one, for primitive integer-sided triangle.
Note that side b is never the smallest side of the triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A353618.
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LINKS
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Table of n, a(n) for n=1..51.
The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.
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FORMULA
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a(n) = A353618(n, 2).
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EXAMPLE
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According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the largest side b = 10 of the first triple (3, 10, 8).
c < a < b with the largest side b = 48 of the 2nd triple (35, 48, 27).
c < b < a with the middle side b = 510 of the 8th triple (539, 510, 216), the first of this type.
The first side b for which there exist two distinct triangles with B = 3*C is for a(11) = a(12) = 840, and these sides b belong respectively to triples (923, 840, 343) and (533, 840, 512).
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MAPLE
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for b from 4 to 9000 do
for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a, b, q)=1 and (b-q^3) < a and a < b+q^3 then print(b); end if;
end do;
end do;
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CROSSREFS
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Cf. A353618 (triples), A353619 (side a), this sequence (side b), A353621 (side c), A353622 (perimeter).
Cf. A343065 (similar, but with B = 2*C).
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KEYWORD
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nonn,new
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AUTHOR
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Bernard Schott, May 07 2022
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STATUS
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approved
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