A338569 -id:A338569 - cp's OEIS frontend

A338568 Number of integers less than n with the same number of unordered factorizations as n.

0, 1, 2, 0, 3, 1, 4, 0, 2, 3, 5, 0, 6, 4, 5, 0, 7, 1, 8, 2, 6, 7, 9, 0, 8, 9, 1, 3, 10, 1, 11, 1, 10, 11, 12, 0, 12, 13, 14, 2, 13, 2, 14, 4, 5, 15, 15, 0, 16, 6, 17, 7, 16, 3, 18, 4, 19, 20, 17, 0, 18, 21, 8, 1, 22, 3, 19, 9, 23, 4, 20, 0, 21, 24, 10, 11, 25, 5, 22, 1

Offset: 1

Ilya Gutkovskiy, Nov 02 2020

nonn

			a(10) = 3 because A001055(10) = 2 and also A001055(4) = A001055(6) = A001055(9) = 2.

Eric Weisstein's World of Mathematics, Unordered Factorization

Cf. A001055, A330972 (positions of 0's), A338569.

T[, 1] = T[1, ] = 1; T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] T[n/#, #] &]; A001055[n_] := T[n, n]; Table[Length[Select[Range[n - 1], A001055[#] == A001055[n] &]], {n, 80}]

a(n) = |{j < n : A001055(j) = A001055(n)}|.

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Nov 03 2020

nonn

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

Eric Weisstein's World of Mathematics, Abelian Group

Cf. A000688, A338568, A338569.

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

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

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

cp's OEIS Frontend

A338568 Number of integers less than n with the same number of unordered factorizations as n.

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