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

1, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23, 245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385, 41, 3125, 119, 115, 143, 1225, 43, 145, 133, 1375, 47, 455, 53, 425, 539, 155, 59, 4375, 169, 605, 161, 475, 61, 1715, 187, 1625, 203

Offset: 1

Author

Gus Wiseman, Oct 28 2022

Keywords

• nonn
• mult

Comments

This is the same as A045966 except the first term is 1 instead of 3.

Examples

			The terms together with their prime indices begin:
    1: {}
    5: {3}
    7: {4}
   25: {3,3}
   11: {5}
   35: {3,4}
   13: {6}
  125: {3,3,3}
   49: {4,4}
   55: {3,5}
   17: {7}
  175: {3,3,4}
   19: {8}
   65: {3,6}
   77: {4,5}
  625: {3,3,3,3}
		

Crossrefs

Applying the transformation only once gives A003961.

A permutation of A007310.

Other multiplicative sequences: A064988, A064989, A357977, A357980, A357983.

A000040 lists the primes.

A056239 adds up prime indices, row-sums of A112798.

Cf. A000720, A003964, A066207, A076610, A215366, A296150, A299201, A357979.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Product[Prime[i+2],{i,primeMS[n]}],{n,30}]

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

Python

from math import prod
from sympy import nextprime, factorint
def A357852(n): return prod(nextprime(p,ith=2)**e for p, e in factorint(n).items()) # Chai Wah Wu, Oct 29 2022

Formula

a(n) = A003961(A003961(n)).