A368728 - OEIS

%I #6 Jan 08 2024 14:30:37

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26,27,

%T 28,29,30,31,32,33,34,35,36,39,40,41,42,43,44,45,46,47,48,49,50,51,52,

%U 54,55,56,58,59,60,62,63,64,65,66,67,68,69,70,72,73,75

%N Numbers whose prime indices are 1, prime, or semiprime.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%F Closed under multiplication.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Max@@Length/@prix/@prix[#]<=2&]

%Y These are products of primes indexed by elements of A037143.

%Y For just primes we have A076610, strict A302590.

%Y For just semiprimes we have A339112, strict A340020.

%Y For squarefree semiprimes we have A339113, strict A309356.

%Y The odd case is A368729, strict A340019.

%Y The complement is A368833.

%Y A000607 counts partitions into primes, A034891 with ones allowed.

%Y A001358 lists semiprimes, squarefree A006881.

%Y A006450, A106349, A322551, A368732 list selected primes.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A101048 counts partitions into semiprimes.

%Y Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628, A320911, A320912.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 07 2024