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

1, 3, 7, 9, 19, 21, 53, 27, 49, 57, 131, 63, 311, 159, 133, 81, 719, 147, 1619, 171, 371, 393, 3671, 189, 361, 933, 343, 477, 8161, 399, 17863, 243, 917, 2157, 1007, 441, 38873, 4857, 2177, 513, 84017, 1113, 180503, 1179, 931, 11013, 386093, 567, 2809, 1083

Offset: 1

Author

Gus Wiseman, May 17 2022

Keywords

• nonn
• mult

Examples

			The terms together with their prime indices begin:
      1: {}
      3: {2}
      7: {4}
      9: {2,2}
     19: {8}
     21: {2,4}
     53: {16}
     27: {2,2,2}
     49: {4,4}
     57: {2,8}
    131: {32}
     63: {2,2,4}
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..397 (calculated using the b-file at A033844)

Crossrefs

These are the positions of first appearances in A353394.

A001222 counts prime factors with multiplicity, distinct A001221.

A003963 gives product of prime indices, counted by A339095.

A033844 lists primes indexed by powers of 2.

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914, product A005361.

A181819 gives prime shadow, firsts A181821, relatively prime A325131.

Equivalent sequence with prime(2*k) instead of prime(2^k): A297002.

Cf. A003586, A005117, A130091, A182850, A289509, A324850, A325755, A353393, A353395, A353399.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@(2^primeMS[n]),{n,100}]

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

Python

from math import prod
from sympy import prime, primepi, factorint
def A353397(n): return prod(prime(2**primepi(p))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 20 2022

Formula

If n = prime(e_1)...prime(e_k), then a(n) = prime(2^(e_1))...prime(2^(e_k)).

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2^k)) = 1.90812936178871496289... . - Amiram Eldar, Dec 09 2022