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

Author

Gus Wiseman, May 19 2019

Keywords

• nonn
• mult

Comments

The union is A308299.

Examples

			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}.
		

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.

Crossrefs

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

Programs

Mathematica

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

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

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

Extensions

Keyword:mult added by Antti Karttunen, Nov 17 2019