A325709 Replace k with k! in the prime indices of n.

1, 2, 3, 4, 13, 6, 89, 8, 9, 26, 659, 12, 5443, 178, 39, 16, 49033, 18, 484037, 52, 267, 1318, 5222429, 24, 169, 10886, 27, 356, 61194647, 78, 774825383, 32, 1977, 98066, 1157, 36, 10552185239, 968074, 16329, 104, 153903050137, 534, 2394322471421, 2636, 117

Offset

1,2

Comments

The union is A308299.

Links

Amiram Eldar, Table of n, a(n) for n = 1..78 (calculated using the b-file at A062439)

Index entries for sequences computed from indices in prime factorization.

Formula

Completely multiplicative with a(prime(n)) = prime(n!).

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - Amiram Eldar, Dec 09 2022

Example

The sequence of terms together with their prime indices begins:
       1: {}
       2: {1}
       3: {2}
       4: {1,1}
      13: {6}
       6: {1,2}
      89: {24}
       8: {1,1,1}
       9: {2,2}
      26: {1,6}
     659: {120}
      12: {1,1,2}
    5443: {720}
     178: {1,24}
      39: {2,6}
      16: {1,1,1,1}
   49033: {5040}
      18: {1,2,2}
  484037: {40320}
      52: {1,1,6}.

Mathematica

Table[Times@@Prime/@(If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]!), {n, 20}]

Prog

(PARI) A325709(n) = { my(f=factor(n)); prod(i=1, #f~, prime(primepi(f[i, 1])!)^f[i, 2]); }; \\ Antti Karttunen, Nov 17 2019
(Python)
from math import prod, factorial
from sympy import prime, primepi, factorint
def A325709(n): return prod(prime(factorial(primepi(p)))**e for p, e in factorint(n).items()) # Chai Wah Wu, Dec 26 2022

Crossrefs

Cf. A000142, A056239, A062439, A064986, A076934, A112798, A115944, A284605, A308299, A322583, A325509, A325616, A325618, A325704.

Sequence in context: A117342 A328210 A303385 * A338478 A126129 A102947

Adjacent sequences: A325706 A325707 A325708 * A325710 A325711 A325712

Keyword

nonn,mult

Author

Gus Wiseman, May 19 2019

Extensions

Keyword:mult added by Antti Karttunen, Nov 17 2019

Status

approved