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%I #37 Mar 24 2023 14:54:23
%S 0,1,1,0,2,1,2,1,1,1,1,2,2,2,2,2,1,2,2,3,2,2,1,3,1,2,1,2,1,5,2,3,1,3,
%T 1,4,1,1,3,4,2,5,2,3,2,3,1,6,2,4,0,3,2,6,1,5,1,3,1,6,2,3,3,6,1,6,1,2,
%U 1,5,1,8,3,4,3,5,1,7,1,6,1,4,1,8,1,5,0,5,2,9,2,4,1,4,0,9,1,3,2,6,1,8,2,7,4
%N a(n) = number of m such that sum of proper divisors of m (A001065(m)) is n.
%C The offset is 2 since there are infinitely many numbers (all the primes) for which A001065 = 1.
%C The graph of this sequence, shifted by 1, looks similar to that of A061358, which counts Goldbach partitions of n. - _T. D. Noe_, Dec 05 2008
%C For n > 2, a(n) <= A000009(n) as all divisor lists must have distinct values. - _Roderick MacPhee_, Sep 13 2016
%C The smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k is A125601(n). - _Bernard Schott_, Mar 23 2023
%H T. D. Noe, <a href="/A048138/b048138.txt">Table of n, a(n) for n = 2..10000</a>
%H Carl Pomerance, <a href="https://www.semanticscholar.org/paper/THE-FIRST-FUNCTION-AND-ITS-ITERATES-Pomerance/b9ac7d56efb36ca0f44defc231c5b267b4779c97">The first function and its iterates</a>, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
%F From _Bernard Schott_, Mar 23 2023: (Start)
%F a(n) = 0 iff n is in A005114 (untouchable numbers).
%F a(n) = 1 iff n is in A057709 ("hermit" numbers).
%F a(n) = 2 iff n is in A057710.
%F a(n) > 1 iff n is in A160133. (End)
%e a(6) = 2 since 6 is the sum of the proper divisors of 6 and 25.
%p with(numtheory): for n from 2 to 150 do count := 0: for m from 1 to n^2 do if sigma(m) - m = n then count := count+1 fi: od: printf(`%d,`,count): od:
%o (PARI) list(n)=my(v=vector(n-1),k); for(m=4,n^2, k=sigma(m)-m; if(k>1 & k<=n, v[k-1]++)); v \\ _Charles R Greathouse IV_, Apr 21 2011
%Y Cf. A001065, A064440, A125601, A238895, A238896 (records).
%Y Cf. A005114, A057709, A057710, A160133.
%K easy,nonn
%O 2,5
%A _Naohiro Nomoto_
%E More terms from _James A. Sellers_, Feb 19 2001
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