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%I #22 Dec 09 2022 07:06:51
%S 1,3,7,9,19,21,53,27,49,57,131,63,311,159,133,81,719,147,1619,171,371,
%T 393,3671,189,361,933,343,477,8161,399,17863,243,917,2157,1007,441,
%U 38873,4857,2177,513,84017,1113,180503,1179,931,11013,386093,567,2809,1083
%N Replace prime(k) with prime(2^k) in the prime factorization of n.
%H Amiram Eldar, <a href="/A353397/b353397.txt">Table of n, a(n) for n = 1..397</a> (calculated using the b-file at A033844)
%F If n = prime(e_1)...prime(e_k), then a(n) = prime(2^(e_1))...prime(2^(e_k)).
%F Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2^k)) = 1.90812936178871496289... . - _Amiram Eldar_, Dec 09 2022
%e The terms together with their prime indices begin:
%e 1: {}
%e 3: {2}
%e 7: {4}
%e 9: {2,2}
%e 19: {8}
%e 21: {2,4}
%e 53: {16}
%e 27: {2,2,2}
%e 49: {4,4}
%e 57: {2,8}
%e 131: {32}
%e 63: {2,2,4}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Times@@Prime/@(2^primeMS[n]),{n,100}]
%o (PARI) a(n) = my(f=factor(n)); for(k=1, #f~, f[k,1] = prime(2^primepi(f[k,1]))); factorback(f); \\ _Michel Marcus_, May 20 2022
%o (Python)
%o from math import prod
%o from sympy import prime, primepi, factorint
%o def A353397(n): return prod(prime(2**primepi(p))**e for p, e in factorint(n).items()) # _Chai Wah Wu_, May 20 2022
%Y These are the positions of first appearances in A353394.
%Y A001222 counts prime factors with multiplicity, distinct A001221.
%Y A003963 gives product of prime indices, counted by A339095.
%Y A033844 lists primes indexed by powers of 2.
%Y A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A124010 gives prime signature, sorted A118914, product A005361.
%Y A181819 gives prime shadow, firsts A181821, relatively prime A325131.
%Y Equivalent sequence with prime(2*k) instead of prime(2^k): A297002.
%Y Cf. A003586, A005117, A130091, A182850, A289509, A324850, A325755, A353393, A353395, A353399.
%K nonn,mult
%O 1,2
%A _Gus Wiseman_, May 17 2022
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