A331973 a(n) is the number of values of m such that the sum of proper infinitary divisors of m (A126168) is n.

0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 1, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 4, 1, 4, 3, 4, 3, 5, 0, 3, 2, 4, 3, 5, 1, 4, 3, 4, 2, 6, 2, 5, 2, 5, 3, 7, 1, 6, 2, 4, 2, 7, 1, 5, 4, 5, 3, 8, 0, 5, 2, 6, 1, 8, 2, 5, 4, 6, 4, 9, 0, 6, 1, 5, 3, 10, 2, 8, 2

Offset: 2

Author

Amiram Eldar, Feb 03 2020

Keywords

• nonn

Comments

The infinitary version of A048138.

The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A126168(k) = 1.

Examples

			a(8) = 2 since 8 is the sum of the proper infinitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..30000

Crossrefs

Cf. A048138, A049417, A077609, A126168, A324938, A331971.

Programs

Mathematica

fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ (fun @@@ FactorInteger[n]); is[n_] := isigma[n] - n; m = 300; v = Table[0, {m}]; Do[i = is[k]; If[2 <= i <= m, v[[i]]++], {k, 1, m^2}]; Rest@v