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A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

%I A353393 #15 Mar 10 2025 12:26:10
%S A353393 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 A353393 83,89,97,101,103,107,109,113,125,127,131,137,139,149,151,157,163,167,
%U A353393 173,179,181,191,193,197,199,211,223,225,227,229,233,239,241,251
%N A353393 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 A353393 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.
%H A353393 Robert Israel, <a href="/A353393/b353393.txt">Table of n, a(n) for n = 1..10000</a>
%F A353393 Equals A353389 U A000040.
%e A353393 The terms together with their prime indices begin:
%e A353393     2: {1}
%e A353393     3: {2}
%e A353393     5: {3}
%e A353393     7: {4}
%e A353393     9: {2,2}
%e A353393    11: {5}
%e A353393    13: {6}
%e A353393    17: {7}
%e A353393    19: {8}
%e A353393    23: {9}
%e A353393    29: {10}
%e A353393    31: {11}
%e A353393    36: {1,1,2,2}
%p A353393 pshadow:= proc(n) local F,i;
%p A353393   F:= ifactors(n)[2];
%p A353393   mul(ithprime(i),i=F[..,2])
%p A353393 end proc:
%p A353393 filter:= proc(n) local s;
%p A353393   if isprime(n) then return true fi;
%p A353393   s:= pshadow(n);
%p A353393   n mod s = 0 and member(s,R)
%p A353393 end proc:
%p A353393 R:= {}:
%p A353393 for i from 2 to 2000 do if filter(i) then R:= R union {i} fi od:
%p A353393 sort(convert(R,list)); # _Robert Israel_, Mar 10 2025
%t A353393 red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
%t A353393 suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
%t A353393 Select[Range[2,200],suQ[#]&]
%Y A353393 The first term that is not a prime power A000961 is 36.
%Y A353393 The first term that is not a prime or a perfect power A001597 is 1260. - Corrected by _Robert Israel_, Mar 10 2025
%Y A353393 The non-recursive version is A325755, counted by A325702.
%Y A353393 Removing all primes gives A353389.
%Y A353393 These partitions are counted by A353426.
%Y A353393 The version for compositions is A353431.
%Y A353393 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353393 A003963 gives product of prime indices.
%Y A353393 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353393 A124010 gives prime signature, sorted A118914.
%Y A353393 A130091 lists numbers with all distinct prime exponents, counted by A098859.
%Y A353393 A181819 gives prime shadow, with an inverse A181821.
%Y A353393 A325131 lists numbers relatively prime to their prime shadow.
%Y A353393 Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.
%K A353393 nonn
%O A353393 1,1
%A A353393 _Gus Wiseman_, May 15 2022