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A355746 Number of different multisets that can be obtained by choosing a prime index (or a prime factor) of each integer from 2 to n.

%I A355746 #16 Aug 03 2022 12:40:02
%S A355746 1,1,1,1,1,2,2,2,2,4,4,6,6,12,20,20,20,26,26,36,58,116,116,140,140,
%T A355746 280,280,384,384,536,536,536,844,1688,2380,2716,2716,5432,8484,10152,
%U A355746 10152,13308,13308,18064,21616,43232,43232,47648,47648,54656,84480,114304,114304
%N A355746 Number of different multisets that can be obtained by choosing a prime index (or a prime factor) of each integer from 2 to n.
%H A355746 Michael S. Branicky, <a href="/A355746/b355746.txt">Table of n, a(n) for n = 1..75</a>
%F A355746 a(n) = A355744(A070826(n)).
%F A355746 a(p) = a(p-1) for p prime. - _Michael S. Branicky_, Aug 03 2022
%e A355746 The a(n) multisets for n = 2, 6, 10, 12:
%e A355746   {1}  {1,1,1,2,3}  {1,1,1,1,1,2,2,3,4}  {1,1,1,1,1,1,2,2,3,4,5}
%e A355746        {1,1,2,2,3}  {1,1,1,1,2,2,2,3,4}  {1,1,1,1,1,2,2,2,3,4,5}
%e A355746                     {1,1,1,1,2,2,3,3,4}  {1,1,1,1,1,2,2,3,3,4,5}
%e A355746                     {1,1,1,2,2,2,3,3,4}  {1,1,1,1,2,2,2,2,3,4,5}
%e A355746                                          {1,1,1,1,2,2,2,3,3,4,5}
%e A355746                                          {1,1,1,2,2,2,2,3,3,4,5}
%t A355746 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355746 Table[Length[Union[Sort/@Tuples[primeMS/@Range[2,n]]]],{n,15}]
%o A355746 (Python)
%o A355746 from sympy import factorint
%o A355746 from itertools import count, islice
%o A355746 def agen():
%o A355746     s = {(1,)}
%o A355746     for n in count(2):
%o A355746         yield len(s)
%o A355746         s = set(tuple(sorted(t+(d,))) for t in s for d in factorint(n))
%o A355746 print(list(islice(agen(), 53))) # _Michael S. Branicky_, Aug 03 2022
%Y A355746 The sum of the same integers is A000096.
%Y A355746 The product of the same integers is A000142, Heinz number A070826.
%Y A355746 The integers themselves are the rows of A131818 (shifted).
%Y A355746 Counting sequences instead of multisets: A355537, with multiplicity A327486.
%Y A355746 Using prime indices instead of 2..n gives A355744, for sequences A355741.
%Y A355746 The version for divisors instead of prime factors is A355747.
%Y A355746 A000040 lists the prime numbers.
%Y A355746 A001221 counts distinct prime factors, with sum A001414.
%Y A355746 A001222 counts prime factors with multiplicity.
%Y A355746 A003963 multiplies together the prime indices of n.
%Y A355746 A056239 adds up prime indices, row sums of A112798.
%Y A355746 Cf. A000720, A002110, A076610, A355538, A355731, A355733, A355740, A355742, A355745.
%K A355746 nonn
%O A355746 1,6
%A A355746 _Gus Wiseman_, Jul 20 2022
%E A355746 a(28) and beyond from _Michael S. Branicky_, Aug 03 2022