A045779 Number of factorizations of n into distinct factors for some n (image of A045778).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 21, 22, 25, 27, 31, 32, 33, 34, 38, 40, 42, 43, 44, 46, 52, 54, 55, 56, 57, 59, 61, 64, 67, 70, 74, 76, 80, 83, 88, 89, 91, 93, 100, 104, 110, 111, 112, 116, 117, 120, 122, 123, 132, 137, 140, 141, 142, 143, 148

Offset

1,2

Comments

We may use A045778(k*m) >= A045778(k) for any k, m >= 1 to disprove presence of some positive integer in this sequence. - David A. Corneth, Oct 24 2024

Links

David A. Corneth, Table of n, a(n) for n = 1..953 (terms <= 10^5)

Wikipedia, Multiplicative partition

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Example

From David A. Corneth, Oct 24 2024: (Start)
5 is a term as 24 has five factorizations into distinct divisors of 24 namely 24 = 2 * 12 = 3 * 8 = 4 * 6 = 2 * 3 * 4 which is five such factorizations.
11 is not a term. From terms in A025487 only the numbers 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 128, 256, 512, 1024 have no more than 11 such factorizations. Any multiple of these numbers in A025487 that is not already listed has more than 11 such factorizations which proves 11 is not in this sequence. (End)

Crossrefs

Factorizations are A001055, with image A045782, with complement A330976.

Strict factorizations are A045778 with image A045779 and complement A330975.

The least number with A045779(n) strict factorizations is A045780(n).

The least number with n strict factorizations is A330974(n).

Cf. A001222, A025487, A033833, A045783, A318286, A328966, A330972, A330973, A330997.

Sequence in context: A064817 A366378 A108549 * A235028 A062014 A164707

Adjacent sequences: A045776 A045777 A045778 * A045780 A045781 A045782

Keyword

nonn

Author

David W. Wilson

Extensions

Name edited by Gus Wiseman, Jan 11 2020

Status

approved