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A325709
Replace k with k! in the prime indices of n.
4
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.
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
KEYWORD
nonn,mult
AUTHOR
Gus Wiseman, May 19 2019
EXTENSIONS
Keyword:mult added by Antti Karttunen, Nov 17 2019
STATUS
approved