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

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, 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, 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

Offset: 2

Author

Naohiro Nomoto

Keywords

• easy
• nonn

Comments

The offset is 2 since there are infinitely many numbers (all the primes) for which A001065 = 1.

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

For n > 2, a(n) <= A000009(n) as all divisor lists must have distinct values. - Roderick MacPhee, Sep 13 2016

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

Examples

			a(6) = 2 since 6 is the sum of the proper divisors of 6 and 25.
		

Links

• T. D. Noe, Table of n, a(n) for n = 2..10000
• Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.

Crossrefs

Cf. A001065, A064440, A125601, A238895, A238896 (records).

Cf. A005114, A057709, A057710, A160133.

Programs

Maple

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:

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

Formula

From Bernard Schott, Mar 23 2023: (Start)

a(n) = 0 iff n is in A005114 (untouchable numbers).

a(n) = 1 iff n is in A057709 ("hermit" numbers).

a(n) = 2 iff n is in A057710.

a(n) > 1 iff n is in A160133. (End)

Extensions

More terms from James Sellers, Feb 19 2001