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 A054973 Number of numbers whose divisors sum to n. 34

%I

%S 1,0,1,1,0,1,1,1,0,0,0,2,1,1,1,0,0,2,0,1,0,0,0,3,0,0,0,1,0,1,2,2,0,0,

%T 0,1,0,1,1,1,0,3,0,1,0,0,0,3,0,0,0,0,0,2,0,2,1,0,0,3,0,1,1,0,0,0,0,1,

%U 0,0,0,5,0,1,0,0,0,1,0,2,0,0,0,3,0,0,0,0,0,3,1,0,1,0,0,4,0

%N Number of numbers whose divisors sum to n.

%C a(n) = frequency of values n in A000203(m), where A000203(m) = sum of divisors of m. a(n) >= 1 for such n that A175192(n) = 1, a(n) >= 1 if A000203(m) = n for any m. a(n) = 0 for such n that A175192(n) = 0, a(n) = 0 if A000203(m) = n has no solution. - _Jaroslav Krizek_, Mar 01 2010

%C First occurrence of k: 2, 1, 12, 24, 96, 72, ..., = A007368. - _Robert G. Wilson v_, May 14 2014

%C a(n) is also the number of positive terms in the n-th row of triangle A299762. - _Omar E. Pol_, Mar 14 2018

%H T. D. Noe, <a href="/A054973/b054973.txt">Table of n, a(n) for n = 1..10000</a>

%e a(12)=2 since 11 has factors 1 and 11 with 1+11=12 and 6 has factors 1, 2, 3 and 6 with 1+2+3+6=12.

%t nn = 105; t = Table[0, {nn}]; k = 1; While[k < 6 nn^(3/2)/Pi^2, d = DivisorSigma[1, k]; If[d < nn + 1, t[[d]]++]; k++]; t (* _Robert G. Wilson v_, May 14 2014 *)

%o (PARI) a(n)=v = vector(0); for (i = 1, n, if (sigma(i) == n, v = concat(v, i));); #v; \\ _Michel Marcus_, Oct 22 2013

%o (PARI) a(n)=sum(k=1,n,sigma(k)==n) \\ _Charles R Greathouse IV_, Nov 12 2013

%o (PARI) first(n)=my(v=vector(n),t); for(k=1,n, t=sigma(n); if(t<=n, v[t]++)); v \\ _Charles R Greathouse IV_, Mar 08 2017

%Y An inverse to the sum-of-divisors function A000203,

%Y For partial sums see A074753.

%Y Cf. A002191, A007609.

%K nonn

%O 1,12

%A _Henry Bottomley_, May 16 2000

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Last modified April 19 04:19 EDT 2019. Contains 322237 sequences. (Running on oeis4.)