0,6

The list of finite subsets of the positive integers in Gray order begins: {}, {1}, {1,2}, {2}, {2,3}, {1,2,3}, {1,3}, {3}, ... cf. A003188, A227738, A360287.

The descent set of permutation p of [n] is the set of indices i with p(i)>p(i+1), a subset of [n-1].

Alois P. Heinz, Rows n = 0..15, flattened

Wikipedia, Gray code

Wikipedia, Permutation

T(5,5) = 4: there are 4 permutations of [5] with descent set {1,2,3} (the 5-th subset in Gray order): 43215, 53214, 54213, 54312.

Triangle T(n,k) begins:

1;

1;

1, 1;

1, 2, 1, 2;

1, 3, 3, 5, 3, 1, 5, 3;

1, 4, 6, 9, 11, 4, 16, 9, 6, 9, 1, 4, 16, 9, 11, 4;

...

a:= proc(n) a(n):= `if`(n<2, n, Bits[Xor](n, a(iquo(n, 2)))) end:

b:= proc(u, o, t) option remember; `if`(u+o=0, x^a(t),

add(b(u-j, o+j-1, t), j=1..u)+

add(b(u+j-1, o-j, t+2^(o+u-1)), j=1..o))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):

seq(T(n), n=0..7);

Row sums give A000142.

Row lengths are A011782.

See A060351, A335845 for similar triangles.

Cf. A003188, A006068, A227738, A360287.

nonn,tabf,new

Alois P. Heinz, Feb 03 2023

approved

0,6

This has the same recurrence as Somos-5 (A006721) with different initial values.

The elliptic curve y^2 + xy = x^3 + x^2 - 2x (LMFDB label 102.a1) has infinite order point P = (2, 2). The x and y coordinates of n*P have denominators a(n)^2 and |a(n)^3| respectively.

If b(2*n) = 6^(1/4)*a(2*n), b(2*n+1) = a(2*n+1), then b(n) is a generalized Somos-4 sequence with b(n+2)*b(n-2) = 6^(1/2)*b(n+1)*b(n-1) - b(n)*b(n) for all n in Z.

This is the sequence T_n in the Hone 2022 paper.

Table of n, a(n) for n=0..25.

A. N. W. Hone, Heron triangles with two rational medians and Somos-5 sequences, European Journal of Mathematics, 8 (2022), 1424-1486; arXiv:2107.03197 [math.NT], 2021-2022.

LMFDB, Elliptic Curve 102.a1 (Cremona label 102a1)

a(2*n) = -A241595(n+1), a(n) = -a(-n) for all n in Z.

5*P = (50/49, 20/343) and a(5) = -7, 6*P = (121/64, -1881/512) and a(6) = 8.

a[0] = 0; a[1] = a[3] = a[4] = 1; a[2] = -1; a[5] = -7;

a[n_?Negative] := -a[-n];

a[n_] := a[n] = (a[n-1] a[n-4] + a[n-2] a[n-3]) / a[n-5]; (* Andrey Zabolotskiy, Feb 05 2023 *)

a[ n_] := Module[{A = Table[1, Max[5, Abs[n]]]}, A[[2]] = -1; A[[5]] = -7; Do[ A[[k]] = (A[[k-1]]*A[[k-4]] + A[[k-2]]*A[[k-3]])/A[[k-5]], {k, 6, Length[A]}]; If[n==0, 0, Sign[n]*A[[Abs[n]]] ]];

(PARI) {a(n) = my(A = vector(max(5, abs(n)), k, 1)); A[2] = -1; A[5] = -7; for(k=6, #A, A[k] = (A[k-1]*A[k-4] + A[k-2]*A[k-3])/A[k-5]); if(n==0, 0, sign(n)*A[abs(n)])};

Cf. A006721, A241595.

sign,new

Michael Somos, Feb 04 2023

approved

1,3

The main diagonal of the diagram called "symmetric representation of sigma(n)" is its axis of symmetry. In this case it is also the first diagonal of the diagram. The second diagonals are the two diagonals that are adjacent to the main diagonal. The third diagonals are the two diagonals that are adjacent to the second diagonals. And so on.

If and only if n is a power of 2 (A000079) then row n lists the first n terms of A040000 (the same sequence as the right border of the triangle).

If and only if n is an odd prime (A065091) then row n lists (n - 1)/2 zeros together with 1 + (n - 1)/2 2's.

If and only if n is an even perfect number (Cf. A000396) then row n lists n 2's (the first n terms of A007395).

For further information about the mentioned "widths" see A249351.

Table of n, a(n) for n=1..105.

Omar E. Pol, Illustration of initial terms of the column 1 = A067742

Index entries for sequences related to sigma(n)

T(n,1) = A067742(n) = A249351(n,n).

T(n,k) = 2*A249351(n,n+k-1), if 1 < k <= n.

Triangle begins (rows: 1..16):

1;

1, 2;

0, 2, 2;

1, 2, 2, 2;

0, 0, 2, 2, 2;

2, 2, 2, 2, 2, 2;

0, 0, 0, 2, 2, 2, 2;

1, 2, 2, 2, 2, 2, 2, 2;

1, 2, 0, 0, 2, 2, 2, 2, 2;

0, 2, 2, 2, 2, 2, 2, 2, 2, 2;

0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2;

2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2;

0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2;

0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;

2, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2;

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;

...

Row sums give A000203.

Column 1 gives A067742.

Right border gives A040000.

Cf. A000079, A000396, A007395, A065091, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A262626.

nonn,tabl,new

Omar E. Pol, Jan 22 2023

approved

0,3

Table of n, a(n) for n=0..11.

Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.

Column k=8 of A333706.

Cf. A110128.

nonn,more,hard,new

Seiichi Manyama, Feb 05 2023

approved

0,3

Table of n, a(n) for n=0..11.

Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.

Column k=7 of A333706.

Cf. A110128.

nonn,hard,more,new

Seiichi Manyama, Feb 05 2023

approved

1,1

Table of n, a(n) for n=1..39.

69 is a term since A360331(69) = A360331(70) = 24.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq

(PARI) s(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))); }

lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s1 == s2, print1(n-1, ", ")); s1 = s2); }

Cf. A360331.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A360358.

nonn,new

Amiram Eldar, Feb 04 2023

approved

1,1

Numbers k such that A360327(k) = A360327(k+1) = 1 are terms of A360357.

Table of n, a(n) for n=1..37.

714 is a term since A360327(714) = A360327(715) = 72 > 1.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e+1)-1)/(p-1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = s[1]; n = 2; c = 0; While[c < 40, s2 = s[n]; If[s1 == s2 > 1, c++; AppendTo[seq, n - 1]]; s1 = s2; n++]; seq

(PARI) s(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1)); }

lista(nmax) = {my(s1 = s(1), s2); for(n=2, nmax, s2=s(n); if(s2 > 1 && s1 == s2, print1(n-1, ", ")); s1 = s2); }

Cf. A360327, A360357.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A360359.

nonn,new

Amiram Eldar, Feb 04 2023

approved

1,2

There are no 3 consecutive integers that are products of primes of nonprime index since 1 out of 3 consecutive integers is divisible by 3 which is a prime-indexed prime (A006450).

If a Mersenne prime (A000668) is a prime of nonprime index, then it is in this sequence. Of the first 10 Mersenne primes 6 are in this in sequence: A000668(k) for k = 2, 5, 7, 8, 9, 10 (see A059305).

Table of n, a(n) for n=1..55.

7 = prime(4) is a term since 4 is nonprime, 7 + 1 = 8 = prime(1)^3, and 1 is also nonprime.

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], ! PrimeQ[PrimePi[#]] &]; seq = {}; q1 = q[1]; n = 2; c = 0; While[c < 55, q2 = q[n]; If[q1 && q2, c++; AppendTo[seq, n - 1]]; q1 = q2; n++]; seq

(PARI) is(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(isprime(primepi(p[i])), return(0))); 1; }

lista(nmax) = {my(q1 = is(1), q2); for(n = 2, nmax, q2 = is(n); if(q1 && q2, print1(n-1, ", ")); q1 = q2); }

Cf. A000668, A006450, A059305, A320628.

nonn,new

Amiram Eldar, Feb 04 2023

approved

1,1

If m is a term then k*m is a term of A360332 for all k in A320628.

Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.

Table of n, a(n) for n=1..46.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[6000], primQ]

(PARI) isab(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n; }

is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};

Subsequence of A360332.

Cf. A320628.

Similar sequences: A006038, A091191, A249263, A302574, A360355.

nonn,new

Amiram Eldar, Feb 04 2023

approved

1,1

If m is a term then k*m is a term of A360328 for all k in A076610.

Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only prime-indexed prime factors.

Table of n, a(n) for n=1..31.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e + 1) - 1)/(p - 1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[10^6], primQ]

(PARI) isab(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1)) > 2*n; }

is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};

Subsequence of A360328.

Cf. A076610.

Similar sequences: A006038, A091191, A249263, A302574, A360356.

nonn,new

Amiram Eldar, Feb 04 2023

approved