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%I #18 Oct 30 2022 08:58:09
%S 1,5,7,25,11,35,13,125,49,55,17,175,19,65,77,625,23,245,29,275,91,85,
%T 31,875,121,95,343,325,37,385,41,3125,119,115,143,1225,43,145,133,
%U 1375,47,455,53,425,539,155,59,4375,169,605,161,475,61,1715,187,1625,203
%N Replace prime(k) with prime(k+2) in the prime factorization of n.
%C This is the same as A045966 except the first term is 1 instead of 3.
%F a(n) = A003961(A003961(n)).
%e The terms together with their prime indices begin:
%e 1: {}
%e 5: {3}
%e 7: {4}
%e 25: {3,3}
%e 11: {5}
%e 35: {3,4}
%e 13: {6}
%e 125: {3,3,3}
%e 49: {4,4}
%e 55: {3,5}
%e 17: {7}
%e 175: {3,3,4}
%e 19: {8}
%e 65: {3,6}
%e 77: {4,5}
%e 625: {3,3,3,3}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Product[Prime[i+2],{i,primeMS[n]}],{n,30}]
%o (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = nextprime(nextprime(f[k,1]+1)+1)); factorback(f); \\ _Michel Marcus_, Oct 28 2022
%o (Python)
%o from math import prod
%o from sympy import nextprime, factorint
%o def A357852(n): return prod(nextprime(p,ith=2)**e for p, e in factorint(n).items()) # _Chai Wah Wu_, Oct 29 2022
%Y Applying the transformation only once gives A003961.
%Y A permutation of A007310.
%Y Other multiplicative sequences: A064988, A064989, A357977, A357980, A357983.
%Y A000040 lists the primes.
%Y A056239 adds up prime indices, row-sums of A112798.
%Y Cf. A000720, A003964, A066207, A076610, A215366, A296150, A299201, A357979.
%K nonn,mult
%O 1,2
%A _Gus Wiseman_, Oct 28 2022
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