A118138 Sum of factorials of prime factors, with multiplicity.

2, 6, 4, 120, 8, 5040, 6, 12, 122, 39916800, 10, 6227020800, 5042, 126, 8, 355687428096000, 14, 121645100408832000, 124, 5046, 39916802, 25852016738884976640000, 12, 240, 6227020802, 18, 5044

Offset

2,1

Comments

For primes p, a(p) = p!. For powers of primes a(p^k) = k*(p!). For nonsquare semiprimes A006881 = pq, we have a(pq)= p! + q!. For sphenic numbers A007304 = p * q * r we have a(pqr) = p! + q! + r!. See also A008472 the sum of the distinct primes dividing n.

Links

Table of n, a(n) for n=2..28.

Eric Weisstein's World of Mathematics, Factorial Sums.

Formula

a(n) = SUM[p|n] p!. a(n) = SUM[i=1..k] e_i * (p_i)! where n = (p_1^e_1)*(p_2^e_2)*...*(p_k^e_k).

Example

a(6) = 8 because 6 = 2 * 3 and 2! + 3! = 8.
a(12) = 10 because 12 = 2^2 * 3 and 2! + 2! + 3! = 10.
a(30) = 128 because 30 = 2 * 3 * 5 and 2! + 3! + 5! = 128.

Mathematica

Table[Total[Flatten[PadRight[{}, Last[#], First[#]]&/@FactorInteger[ n]]!], {n, 30}] (* Harvey P. Dale, Jan 06 2012 *)

Prog

(PARI) a(n)=my(f=factor(n)); sum(i=1, #f~, f[i, 1]!*f[i, 2]) \\ Charles R Greathouse IV, Sep 14 2015

Crossrefs

Cf. A000040, A000142, A006881, A007304, A008472.

Sequence in context: A022404 A210420 A306585 * A240023 A004583 A348720

Adjacent sequences: A118135 A118136 A118137 * A118139 A118140 A118141

Keyword

easy,nonn

Author

Jonathan Vos Post, May 13 2006

Status

approved