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%I #49 Mar 23 2024 08:22:51
%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
%C 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
%H T. D. Noe, <a href="/A054973/b054973.txt">Table of n, a(n) for n = 1..10000</a>
%H Max Alekseyev, <a href="https://home.gwu.edu/~maxal/gpscripts/invphi.gp">PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp</a>, Oct. 2005.
%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 *)
%t Table[Length[Select[IntegerPartitions[n],#==Reverse[Divisors[Max@@#]]&]],{n,30}] (* _Gus Wiseman_, Mar 22 2024 *)
%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
%o (PARI) A054973(n)=#invsigma(n) \\ See Alekseyev link for invsigma(). - _M. F. Hasler_, Nov 21 2019
%Y Cf. A000203 (sum-of-divisors function). [Incorrect comment deleted by _M. F. Hasler_, Nov 21 2019]
%Y For partial sums see A074753.
%Y Cf. A002191, A007609.
%Y The non-strict version is A371284, ranks A371288.
%Y These partitions have ranks A371283, unsorted version A275700.
%Y A000005 counts divisors, row-lengths of A027750.
%Y A000041 counts integer partitions, strict A000009.
%Y Cf. A001221, A002865, A008289, A371286, A371285.
%K nonn
%O 1,12
%A _Henry Bottomley_, May 16 2000
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