%I #27 Oct 25 2024 09:35:45

%S 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,

%T 40,42,43,44,46,52,54,55,56,57,59,61,64,67,70,74,76,80,83,88,89,91,93,

%U 100,104,110,111,112,116,117,120,122,123,132,137,140,141,142,143,148

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

%C 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

%H David A. Corneth, <a href="/A045779/b045779.txt">Table of n, a(n) for n = 1..953</a> (terms <= 10^5)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiplicative_partition">Multiplicative partition</a>

%H R. E. Canfield, P. Erdős and C. Pomerance, <a href="http://math.dartmouth.edu/~carlp/PDF/paper39.pdf">On a Problem of Oppenheim concerning "Factorisatio Numerorum"</a>, J. Number Theory 17 (1983), 1-28.

%e From _David A. Corneth_, Oct 24 2024: (Start)

%e 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.

%e 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)

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

%Y Strict factorizations are A045778 with image A045779 and complement A330975.

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

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

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

%K nonn

%O 1,2

%A _David W. Wilson_

%E Name edited by _Gus Wiseman_, Jan 11 2020