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

%I A325709 #24 Dec 26 2022 15:32:35
%S A325709 1,2,3,4,13,6,89,8,9,26,659,12,5443,178,39,16,49033,18,484037,52,267,
%T A325709 1318,5222429,24,169,10886,27,356,61194647,78,774825383,32,1977,98066,
%U A325709 1157,36,10552185239,968074,16329,104,153903050137,534,2394322471421,2636,117
%N A325709 Replace k with k! in the prime indices of n.
%C A325709 The union is A308299.
%H A325709 Amiram Eldar, <a href="/A325709/b325709.txt">Table of n, a(n) for n = 1..78</a> (calculated using the b-file at A062439)
%H A325709 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A325709 Completely multiplicative with a(prime(n)) = prime(n!).
%F A325709 Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - _Amiram Eldar_, Dec 09 2022
%e A325709 The sequence of terms together with their prime indices begins:
%e A325709        1: {}
%e A325709        2: {1}
%e A325709        3: {2}
%e A325709        4: {1,1}
%e A325709       13: {6}
%e A325709        6: {1,2}
%e A325709       89: {24}
%e A325709        8: {1,1,1}
%e A325709        9: {2,2}
%e A325709       26: {1,6}
%e A325709      659: {120}
%e A325709       12: {1,1,2}
%e A325709     5443: {720}
%e A325709      178: {1,24}
%e A325709       39: {2,6}
%e A325709       16: {1,1,1,1}
%e A325709    49033: {5040}
%e A325709       18: {1,2,2}
%e A325709   484037: {40320}
%e A325709       52: {1,1,6}.
%t A325709 Table[Times@@Prime/@(If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]!),{n,20}]
%o A325709 (PARI) A325709(n) = { my(f=factor(n)); prod(i=1,#f~,prime(primepi(f[i, 1])!)^f[i, 2]); }; \\ _Antti Karttunen_, Nov 17 2019
%o A325709 (Python)
%o A325709 from math import prod, factorial
%o A325709 from sympy import prime, primepi, factorint
%o A325709 def A325709(n): return prod(prime(factorial(primepi(p)))**e for p, e in factorint(n).items()) # _Chai Wah Wu_, Dec 26 2022
%Y A325709 Cf. A000142, A056239, A062439, A064986, A076934, A112798, A115944, A284605, A308299, A322583, A325509, A325616, A325618, A325704.
%K A325709 nonn,mult
%O A325709 1,2
%A A325709 _Gus Wiseman_, May 19 2019
%E A325709 Keyword:mult added by _Antti Karttunen_, Nov 17 2019