

A054973


Number of numbers whose divisors sum to n.


41



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, 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, 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
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OFFSET

1,12


COMMENTS

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
First occurrence of k: 2, 1, 12, 24, 96, 72, ..., = A007368.  Robert G. Wilson v, May 14 2014
a(n) is also the number of positive terms in the nth row of triangle A299762.  Omar E. Pol, Mar 14 2018


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005


EXAMPLE

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.


MATHEMATICA

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 *)


PROG

(PARI) a(n)=v = vector(0); for (i = 1, n, if (sigma(i) == n, v = concat(v, i)); ); #v; \\ Michel Marcus, Oct 22 2013
(PARI) a(n)=sum(k=1, n, sigma(k)==n) \\ Charles R Greathouse IV, Nov 12 2013
(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
(PARI) A054973(n)=#invsigma(n) \\ See Alekseyev link for invsigma().  M. F. Hasler, Nov 21 2019


CROSSREFS

Cf. A000203 (sumofdivisors function). [Incorrect comment deleted by M. F. Hasler, Nov 21 2019]
For partial sums see A074753.
Cf. A002191, A007609.
Sequence in context: A089053 A214979 A068462 * A030351 A257994 A188921
Adjacent sequences: A054970 A054971 A054972 * A054974 A054975 A054976


KEYWORD

nonn


AUTHOR

Henry Bottomley, May 16 2000


STATUS

approved



