cp's OEIS Frontend

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

%I A330993 #9 Jul 17 2021 11:24:55
%S A330993 3,4,5,7,8,10,11,12,13,21,22,25,33,38,41,45,46,49,50,55,57,58,63
%N A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.
%C A330993 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 A330993 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 A330993 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 A330993 A001055(A181821(a(n))) belongs to A000040.
%e A330993 The multiset partitions for n = 1..6:
%e A330993   {11}    {12}    {111}      {1111}        {123}      {1112}
%e A330993   {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
%e A330993                   {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
%e A330993                              {1}{1}{11}    {3}{12}    {2}{111}
%e A330993                              {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
%e A330993                                                       {1}{2}{11}
%e A330993                                                       {1}{1}{1}{2}
%e A330993 The factorizations for n = 1..8:
%e A330993   4    6    8      16       30     24       32         60
%e A330993   2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
%e A330993             2*2*2  4*4      2*15   4*6      2*16       3*20
%e A330993                    2*2*4    3*10   2*12     2*2*8      4*15
%e A330993                    2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
%e A330993                                    2*3*4    2*2*2*4    6*10
%e A330993                                    2*2*2*3  2*2*2*2*2  2*5*6
%e A330993                                                        3*4*5
%e A330993                                                        2*2*15
%e A330993                                                        2*3*10
%e A330993                                                        2*2*3*5
%t A330993 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A330993 unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A330993 Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]
%Y A330993 The same for powers of 2 (instead of primes) is A330990.
%Y A330993 Factorizations are A001055, with image A045782, with complement A330976.
%Y A330993 Numbers whose number of integer partitions is prime are A046063.
%Y A330993 Numbers whose number of strict integer partitions is prime are A035359.
%Y A330993 Numbers whose number of set partitions is prime are A051130.
%Y A330993 Numbers whose number of factorizations is a power of 2 are A330977.
%Y A330993 The least number with prime(n) factorizations is A330992(n).
%Y A330993 Factorizations of a number's inverse prime shadow are A318284.
%Y A330993 Numbers with a prime number of factorizations are A330991.
%Y A330993 Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.
%K A330993 nonn,more
%O A330993 1,1
%A A330993 _Gus Wiseman_, Jan 07 2020