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

1,2

Links

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

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

Example

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}

Mathematica

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

Prog

(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

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.

Sequence in context: A110674 A003528 A032913 * A261072 A154508 A073573

Adjacent sequences: A353394 A353395 A353396 * A353398 A353399 A353400

Keyword

nonn,mult

Author

Gus Wiseman, May 17 2022

Status

approved