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A048138 a(n) = number of m such that sum of proper divisors of m (A001065(m)) is n. 20


%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

%D Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.

%H T. D. Noe, <a href="/A048138/b048138.txt">Table of n, a(n) for n = 2..10000</a>

%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, A005114, A064440, A238895, A238896 (records).

%K easy,nonn

%O 2,5

%A _Naohiro Nomoto_

%E More terms from _James A. Sellers_, Feb 19 2001

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Last modified October 19 00:36 EDT 2018. Contains 316327 sequences. (Running on oeis4.)