A338572 Number of integers less than n with the same number of factorizations into prime powers as n.

0, 1, 2, 0, 3, 4, 5, 0, 1, 6, 7, 2, 8, 9, 10, 0, 11, 3, 12, 4, 13, 14, 15, 1, 5, 16, 2, 6, 17, 18, 19, 0, 20, 21, 22, 0, 23, 24, 25, 3, 26, 27, 28, 7, 8, 29, 30, 1, 9, 10, 31, 11, 32, 4, 33, 5, 34, 35, 36, 12, 37, 38, 13, 0, 39, 40, 41, 14, 42, 43, 44, 0, 45, 46, 15, 16, 47, 48, 49, 2

Offset: 1

Author

Ilya Gutkovskiy, Nov 03 2020

Keywords

• nonn

Examples

			a(18) = 3 because A000688(18) = 2 and also A000688(4) = A000688(9) = A000688(12) = 2.
		

Links

• Eric Weisstein's World of Mathematics, Abelian Group

Crossrefs

Cf. A000688, A338568, A338569.

Programs

Mathematica

Table[Length[Select[Range[n - 1], FiniteAbelianGroupCount[#] == FiniteAbelianGroupCount[n] &]], {n, 80}]

PARI

nf(n) = my(f=factor(n)[, 2]); prod(i=1, #f, numbpart(f[i])); \\ A000688
a(n) = my(nb=nf(n)); sum(k=1, n-1, nf(k) == nb); \\ Michel Marcus, Nov 03 2020

Formula

a(n) = |{j < n : A000688(j) = A000688(n)}|.