cp's OEIS Frontend

A353394 Product of prime shadows of prime indices of n (with multiplicity).

%I A353394 #5 May 20 2022 08:51:01
%S A353394 1,1,2,1,2,2,3,1,4,2,2,2,4,3,4,1,2,4,5,2,6,2,3,2,4,4,8,3,4,4,2,1,4,2,
%T A353394 6,4,6,5,8,2,2,6,4,2,8,3,4,2,9,4,4,4,7,8,4,3,10,4,2,4,6,2,12,1,8,4,2,
%U A353394 2,6,6,6,4,4,6,8,5,6,8,4,2,16,2,2,6,4,4
%N A353394 Product of prime shadows of prime indices of n (with multiplicity).
%C A353394 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.
%C A353394 We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
%F A353394 a(n) = Product_i A181819(A112798(n,i)).
%F A353394 Positions where a(n) = A003963(n) are A003586.
%F A353394 Positions where a(n) = A005361(n) are A353399, counted by A353398.
%F A353394 Positions where a(n) = A181819(n) are A353395, counted by A353396.
%e A353394 We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.
%t A353394 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A353394 red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
%t A353394 Table[Times@@red/@primeMS[n],{n,100}]
%Y A353394 Positions of first appearances are A353397.
%Y A353394 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353394 A003963 gives product of prime indices.
%Y A353394 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353394 A124010 gives prime signature, sorted A118914, product A005361.
%Y A353394 A181819 gives prime shadow, with an inverse A181821.
%Y A353394 A324850 lists numbers divisible by the product of their prime indices.
%Y A353394 A325131 lists numbers relatively prime to their prime shadow.
%Y A353394 A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
%Y A353394 Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.
%K A353394 nonn
%O A353394 1,3
%A A353394 _Gus Wiseman_, May 17 2022