A330993 - OEIS

%I #9 Jul 17 2021 11:24:55

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

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

%C 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}.

%C 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.

%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.

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

%e The multiset partitions for n = 1..6:

%e   {11}    {12}    {111}      {1111}        {123}      {1112}

%e   {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}

%e                   {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}

%e                              {1}{1}{11}    {3}{12}    {2}{111}

%e                              {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}

%e                                                       {1}{2}{11}

%e                                                       {1}{1}{1}{2}

%e The factorizations for n = 1..8:

%e   4    6    8      16       30     24       32         60

%e   2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30

%e             2*2*2  4*4      2*15   4*6      2*16       3*20

%e                    2*2*4    3*10   2*12     2*2*8      4*15

%e                    2*2*2*2  2*3*5  2*2*6    2*4*4      5*12

%e                                    2*3*4    2*2*2*4    6*10

%e                                    2*2*2*3  2*2*2*2*2  2*5*6

%e                                                        3*4*5

%e                                                        2*2*15

%e                                                        2*3*10

%e                                                        2*2*3*5

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

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

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

%Y Numbers whose number of integer partitions is prime are A046063.

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

%Y Numbers whose number of set partitions is prime are A051130.

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

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

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

%Y Numbers with a prime number of factorizations are A330991.

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

%K nonn,more

%O 1,1

%A _Gus Wiseman_, Jan 07 2020