A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

3, 4, 5, 7, 8, 10, 11, 12, 13, 21, 22, 25, 33, 38, 41, 45, 46, 49, 50, 55, 57, 58, 63

Offset: 1

Author

Gus Wiseman, Jan 07 2020

Keywords

• nonn
• more

Comments

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Also numbers whose inverse prime shadow has a prime number of factorizations. A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798. The inverse prime shadow of k is the least number whose prime exponents are the prime indices of k.

Examples

			The multiset partitions for n = 1..6:
  {11}    {12}    {111}      {1111}        {123}      {1112}
  {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
                  {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
                             {1}{1}{11}    {3}{12}    {2}{111}
                             {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
                                                      {1}{2}{11}
                                                      {1}{1}{1}{2}
The factorizations for n = 1..8:
  4    6    8      16       30     24       32         60
  2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
            2*2*2  4*4      2*15   4*6      2*16       3*20
                   2*2*4    3*10   2*12     2*2*8      4*15
                   2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
                                   2*3*4    2*2*2*4    6*10
                                   2*2*2*3  2*2*2*2*2  2*5*6
                                                       3*4*5
                                                       2*2*15
                                                       2*3*10
                                                       2*2*3*5
		

Links

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

Crossrefs

The same for powers of 2 (instead of primes) is A330990.

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

Numbers whose number of integer partitions is prime are A046063.

Numbers whose number of strict integer partitions is prime are A035359.

Numbers whose number of set partitions is prime are A051130.

Numbers whose number of factorizations is a power of 2 are A330977.

The least number with prime(n) factorizations is A330992(n).

Factorizations of a number's inverse prime shadow are A318284.

Numbers with a prime number of factorizations are A330991.

Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

Formula

A001055(A181821(a(n))) belongs to A000040.