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%I #29 Mar 13 2021 12:43:43
%S 0,1,2,2,6,3,24,3,4,7,120,4,720,25,8,4,5040,5,40320,8,26,121,362880,5,
%T 12,721,6,26,3628800,9,39916800,5,122,5041,30,6,479001600,40321,722,9,
%U 6227020800,27,87178291200,122,10,362881,1307674368000,6,48,13,5042,722,20922789888000,7,126,27,40322,3628801,355687428096000,10,6402373705728000,39916801,28,6,726,123
%N a(1) = 0, a(n) = (e1*i1! + e2*i2! + ... + ez*iz!) for n = prime(i1)^e1 * prime(i2)^e2 * ... * prime(iz)^ez, where prime(k) is the k-th prime, A000040(k).
%C Additive with a(p^e) = e * (PrimePi(p)!), where PrimePi(n) = A000720(n).
%C a(3181) has 1001 decimal digits. - _Michael De Vlieger_, Dec 24 2017
%H Michael De Vlieger, <a href="/A276075/b276075.txt">Table of n, a(n) for n = 1..3180</a> (First 120 terms from Antti Karttunen).
%F a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A000142(A055396(n))).
%F Other identities.
%F For all n >= 0:
%F a(A276076(n)) = n.
%F a(A002110(n)) = A007489(n).
%F a(A019565(n)) = A059590(n).
%F a(A206296(n)) = A276080(n).
%F a(A260443(n)) = A276081(n).
%F For all n >= 1:
%F a(A000040(n)) = n! = A000142(n).
%F a(A076954(n-1)) = A033312(n).
%t Array[If[# == 1, 0, Total[FactorInteger[#] /. {p_, e_} /; p > 1 :> e PrimePi[p]!]] &, 66] (* _Michael De Vlieger_, Dec 24 2017 *)
%o (Scheme, with memoization-macro definec)
%o (definec (A276075 n) (cond ((= 1 n) (- n 1)) (else (+ (* (A067029 n) (A000142 (A055396 n))) (A276075 (A028234 n))))))
%o (Python)
%o from sympy import factorint, factorial as f, primepi
%o def a(n):
%o F=factorint(n)
%o return 0 if n==1 else sum(F[i]*f(primepi(i)) for i in F)
%o print([a(n) for n in range(1, 121)]) # _Indranil Ghosh_, Jun 21 2017
%Y Cf. A000040, A000142, A000720, A002110, A007489, A019565, A028234, A033312, A055396, A059590, A067029, A076954, A206296, A260443, A276080, A276081.
%Y Left inverse of A276076.
%Y Cf. also A048675.
%K nonn
%O 1,3
%A _Antti Karttunen_, Aug 18 2016
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