A330992 Least positive integer with exactly prime(n) factorizations into factors > 1, or 0 if no such integer exists.

4, 8, 16, 24, 60, 0, 0, 96, 0, 144, 216, 0, 0, 0, 288, 0, 0, 0, 768, 0, 0, 0, 0, 0, 864, 8192, 0, 0, 1080, 0, 0, 0, 1800, 3072, 0, 0, 0, 0, 0, 0, 0, 2304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3456, 0, 3600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24576

Offset

1,1

Links

Table of n, a(n) for n=1..74.

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

Example

Factorizations of the initial positive terms are:
  4    8      16       24       60       96
  2*2  2*4    2*8      3*8      2*30     2*48
       2*2*2  4*4      4*6      3*20     3*32
              2*2*4    2*12     4*15     4*24
              2*2*2*2  2*2*6    5*12     6*16
                       2*3*4    6*10     8*12
                       2*2*2*3  2*5*6    2*6*8
                                3*4*5    3*4*8
                                2*2*15   4*4*6
                                2*3*10   2*2*24
                                2*2*3*5  2*3*16
                                         2*4*12
                                         2*2*3*8
                                         2*2*4*6
                                         2*3*4*4
                                         2*2*2*12
                                         2*2*2*2*6
                                         2*2*2*3*4
                                         2*2*2*2*2*3

Crossrefs

All positive terms belong to A025487 and also A033833.

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

Numbers whose number of partitions is prime are A046063.

Numbers whose number of strict partitions is prime are A035359.

Numbers whose number of set partitions is prime are A051130.

Numbers with a prime number of factorizations are A330991.

The least number with exactly 2^n factorizations is A330989(n).

Cf. A001222, A045783, A325238, A330972, A330973, A330976, A330993, A330998.

Sequence in context: A343421 A337353 A368614 * A246067 A161226 A369566

Adjacent sequences: A330989 A330990 A330991 * A330993 A330994 A330995

Keyword

nonn

Author

Gus Wiseman, Jan 07 2020

Extensions

More terms from Jinyuan Wang, Jul 07 2021

Status

approved