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A032741
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a(0) = 0; for n > 0, a(n) = number of proper divisors of n (divisors of n which are less than n).
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182
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0, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 11, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 11, 3, 3, 3, 7, 1, 11, 3, 5, 3, 3, 3, 11, 1, 5, 5
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OFFSET
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0,5
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COMMENTS
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Number of d < n which divide n.
Call an integer k between 1 and n a "semi-divisor" of n if n leaves a remainder of 1 when divided by k, i.e., n == 1 (mod k). a(n) gives the number of semi-divisors of n+1. - Joseph L. Pe, Sep 11 2002
a(n+1) is also the number of k, 0 <= k <= n-1, such that C(n,k) divides C(n,k+1). - Benoit Cloitre, Oct 17 2002
a(n+1) is also the number of factors of the n-th degree polynomial x^n + x^(n-1) + x^(n-2) + ... + x^2 + x + 1. Example: 1 + x + x^2 + x^3 = (1+x)(1+x^2) implies a(4)=2.
a(n) is also the number of factors of the n-th Fibonacci polynomial. - T. D. Noe, Mar 09 2006
Number of partitions of n into 2 parts with the second dividing the first. - Franklin T. Adams-Watters, Sep 20 2006
Number of partitions of n+1 into exactly one q and at least one q+1. Example: a(12)=5; indeed, we have 13 = 7 + 6 = 5 + 4 + 4 = 4 + 3 + 3 + 3 = 3 + 2 + 2 + 2 + 2 + 2 = 2 + 11*1.
Differences of A002541. - George Beck, Feb 12 2012
For n > 1: number of ones in row n+1 of triangle A051778. - Reinhard Zumkeller, Dec 03 2014
For n > 0, a(n) is the number of strong divisors of n. - Omar E. Pol, May 03 2015
a(n) is also the number of factors of the (n-1)-th degree polynomial ((x+1)^n-1)/x. Example: for n=6, ((x+1)^6-1)/x) = x^5 + 6*x^4 + 15*x^3 + 20*x^2 + 15*x + 6 = (2+x)(1+x+x^2)(3+3x+x^2) implies a(6)=3. - Federico Provvedi, Oct 09 2018
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REFERENCES
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André Weil, Number Theory, An approach through history, From Hammurapi to Legendre, Birkhäuser, 1984, page 5.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Proper divisors
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FORMULA
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a(n) = tau(n)-1 = A000005(n)-1. Cf. A039653.
G.f.: Sum_{n>=1} x^(2*n)/(1-x^n). - Michael Somos, Apr 29, 2003
G.f.: Sum_{i>=1} (1-x^i+x^(2*i))/(1-x^i). - Jon Perry, Jul 03 2004
a(n) = Sum_{k=1..floor(n/2)} A051731(n-k,k). - Reinhard Zumkeller, Nov 01 2009
G.f.: 2*Sum_{n>=1} Sum_{d|n} log(1 - x^(n/d))^(2*d) / (2*d)!. - Paul D. Hanna, Aug 21 2014
Dirichlet g.f.: zeta(s)*(zeta(s)-1). - Geoffrey Critzer, Dec 06 2014
a(n) = Sum_{k=1..n-1} binomial((n-1) mod k, k-1). - Wesley Ivan Hurt, Sep 26 2016
a(n) = Sum_{i=1..n-1} floor(n/i)-floor((n-1)/i). - Wesley Ivan Hurt, Nov 15 2017
a(n) = Sum_{i=1..n-1} 1-sign(i mod (n-i)). - Wesley Ivan Hurt, Sep 27 2018
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EXAMPLE
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a(6) = 3 since the proper divisors of 6 are 1, 2, 3.
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MAPLE
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A032741 := proc(n)
if n = 0 then
0 ;
else
numtheory[tau](n)-1 ;
end if;
end proc: # R. J. Mathar, Feb 03 2013
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MATHEMATICA
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Prepend[DivisorSigma[0, Range[99]]-1, 0] (* Jayanta Basu, May 25 2013 *)
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PROG
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(PARI) a(n) = if(n<1, 0, numdiv(n)-1)
(PARI) {a(n)=polcoeff(2*sum(m=1, n\2+1, sumdiv(m, d, log(1-x^(m/d) +x*O(x^n) )^(2*d)/(2*d)!)), n)} \\ Paul D. Hanna, Aug 21 2014
(Haskell)
a032741 n = if n == 0 then 0 else a000005 n - 1
-- Reinhard Zumkeller, Jul 31 2014
(GAP) Concatenation([0], List([1..100], n->Tau(n)-1)); # Muniru A Asiru, Oct 09 2018
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CROSSREFS
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Column 2 of A122934.
Cf. A003238, A001065, A027749, A027751 (list of proper divisors).
Cf. A051778, A070824, A091818.
Sequence in context: A325770 A305611 A325765 * A353338 A319149 A344462
Adjacent sequences: A032738 A032739 A032740 * A032742 A032743 A032744
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KEYWORD
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nonn,easy
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AUTHOR
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Patrick De Geest, May 15 1998
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EXTENSIONS
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Typos in definition corrected by Omar E. Pol, Dec 13 2008
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STATUS
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approved
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