%I #9 May 17 2022 01:49:03

%S 2,3,5,7,9,11,13,17,19,23,29,31,36,37,41,43,47,53,59,61,67,71,73,79,

%T 83,89,97,101,103,107,109,113,125,127,131,137,139,149,151,157,163,167,

%U 173,179,181,191,193,197,199,211,223,225,227,229,233,239,241,251

%N Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

%C 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 Equals A353389 U A000040.

%e The terms together with their prime indices begin:

%e 2: {1}

%e 3: {2}

%e 5: {3}

%e 7: {4}

%e 9: {2,2}

%e 11: {5}

%e 13: {6}

%e 17: {7}

%e 19: {8}

%e 23: {9}

%e 29: {10}

%e 31: {11}

%e 36: {1,1,2,2}

%t red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];

%t suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];

%t Select[Range[2,200],suQ[#]&]

%Y The first term that is not a prime power A000961 is 36.

%Y The first term that is not a perfect power A001597 is 1260.

%Y The non-recursive version is A325755, counted by A325702.

%Y Removing all primes gives A353389.

%Y These partitions are counted by A353426.

%Y The version for compositions is A353431.

%Y A001222 counts prime factors with multiplicity, distinct A001221.

%Y A003963 gives product of prime indices.

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A124010 gives prime signature, sorted A118914.

%Y A130091 lists numbers with all distinct prime exponents, counted by A098859.

%Y A181819 gives prime shadow, with an inverse A181821.

%Y A325131 lists numbers relatively prime to their prime shadow.

%Y Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 15 2022