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

Author

Antti Karttunen, Aug 18 2016

Keywords

• nonn

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).

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.

Programs

Mathematica

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

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

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).