A276075 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).

0, 1, 2, 2, 6, 3, 24, 3, 4, 7, 120, 4, 720, 25, 8, 4, 5040, 5, 40320, 8, 26, 121, 362880, 5, 12, 721, 6, 26, 3628800, 9, 39916800, 5, 122, 5041, 30, 6, 479001600, 40321, 722, 9, 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

Offset

1,3

Comments

Additive with a(p^e) = e * (PrimePi(p)!), where PrimePi(n) = A000720(n).

a(3181) has 1001 decimal digits. - Michael De Vlieger, Dec 24 2017

Links

Michael De Vlieger, Table of n, a(n) for n = 1..3180 (First 120 terms from Antti Karttunen).

Formula

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A000142(A055396(n))).

Other identities.

For all n >= 0:

a(A276076(n)) = n.

a(A002110(n)) = A007489(n).

a(A019565(n)) = A059590(n).

a(A206296(n)) = A276080(n).

a(A260443(n)) = A276081(n).

For all n >= 1:

a(A000040(n)) = n! = A000142(n).

a(A076954(n-1)) = A033312(n).

Mathematica

Array[If[# == 1, 0, Total[FactorInteger[#] /. {p_, e_} /; p > 1 :> e PrimePi[p]!]] &, 66] (* Michael De Vlieger, Dec 24 2017 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A276075 n) (cond ((= 1 n) (- n 1)) (else (+ (* (A067029 n) (A000142 (A055396 n))) (A276075 (A028234 n))))))
(Python)
from sympy import factorint, factorial as f, primepi
def a(n):
    F=factorint(n)
    return 0 if n==1 else sum(F[i]*f(primepi(i)) for i in F)
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Jun 21 2017

Crossrefs

Cf. A000040, A000142, A000720, A002110, A007489, A019565, A028234, A033312, A055396, A059590, A067029, A076954, A206296, A260443, A276080, A276081.

Left inverse of A276076.

Cf. also A048675.

Sequence in context: A266479 A296147 A130712 * A321908 A130728 A372576

Adjacent sequences: A276072 A276073 A276074 * A276076 A276077 A276078

Keyword

nonn

Author

Antti Karttunen, Aug 18 2016

Status

approved