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A118138 Sum of factorials of prime factors, with multiplicity. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A175995

Adjacent sequences:  A118135 A118136 A118137 * A118139 A118140 A118141

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, May 13 2006

STATUS

approved

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Last modified April 19 18:37 EDT 2019. Contains 322290 sequences. (Running on oeis4.)