cp's OEIS Frontend

A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

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