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A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

%I A353389 #9 May 16 2022 17:23:40
%S A353389 9,36,125,225,441,1089,1260,1521,1980,2340,2401,2601,2772,3060,3249,
%T A353389 3276,3420,4140,4284,4761,4788,5148,5220,5580,5796,6660,6732,7308,
%U A353389 7380,7524,7569,7740,7812,7956,8460,8649,8892,9108,9324,9540,10332,10620,10764,10836
%N A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.
%C A353389 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.
%C A353389 Said differently, these are nonprime numbers > 1 whose prime shadow is a divisor that is either a prime number or a number already in the sequence.
%e A353389 The initial terms and their prime indices:
%e A353389      9: {2,2}
%e A353389     36: {1,1,2,2}
%e A353389    125: {3,3,3}
%e A353389    225: {2,2,3,3}
%e A353389    441: {2,2,4,4}
%e A353389   1089: {2,2,5,5}
%e A353389   1260: {1,1,2,2,3,4}
%e A353389   1521: {2,2,6,6}
%e A353389   1980: {1,1,2,2,3,5}
%t A353389 red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
%t A353389 suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
%t A353389 Select[Range[2,2000],suQ[#]&&!PrimeQ[#]&]
%Y A353389 The first term that is not a perfect power A001597 is 1260.
%Y A353389 Without the recursion we have A325755 (a superset), counted by A325702.
%Y A353389 Before removing the primes we had A353393.
%Y A353389 These partitions are counted by A353426 minus one.
%Y A353389 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353389 A003963 gives product of prime indices.
%Y A353389 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353389 A124010 gives prime signature, sorted A118914.
%Y A353389 A181819 gives prime shadow, with an inverse A181821.
%Y A353389 A182850 and A323014 give frequency depth, counted by A225485 and A325280.
%Y A353389 A325131 lists numbers relatively prime to their prime shadow.
%Y A353389 Cf. A000005, A000040, A005117, A316413, A316428, A324850, A353394, A353395, A353397, A353399.
%K A353389 nonn
%O A353389 1,1
%A A353389 _Gus Wiseman_, May 15 2022