A054973 Number of numbers whose divisors sum to n.

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

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 n-th row of triangle A299762. - Omar E. Pol, Mar 14 2018

Also the number of integer partitions of n whose parts form the set of divisors of some number (necessarily the greatest part). The Heinz numbers of these partitions are given by A371283. For example, the a(24) = 3 partitions are: (23,1), (15,5,3,1), (14,7,2,1). - Gus Wiseman, Mar 22 2024

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Max 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 *)
Table[Length[Select[IntegerPartitions[n], #==Reverse[Divisors[Max@@#]]&]], {n, 30}] (* Gus Wiseman, Mar 22 2024 *)

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 (sum-of-divisors function). [Incorrect comment deleted by M. F. Hasler, Nov 21 2019]

For partial sums see A074753.

Cf. A002191, A007609.

The non-strict version is A371284, ranks A371288.

These partitions have ranks A371283, unsorted version A275700.

A000005 counts divisors, row-lengths of A027750.

A000041 counts integer partitions, strict A000009.

Cf. A001221, A002865, A008289, A371286, A371285.

Sequence in context: A214979 A068462 A369596 * A030351 A257994 A188921

Adjacent sequences: A054970 A054971 A054972 * A054974 A054975 A054976

Keyword

nonn

Author

Henry Bottomley, May 16 2000

Status

approved