cp's OEIS Frontend

A353397 Replace prime(k) with prime(2^k) in the prime factorization of n.

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