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A022559 Sum of exponents in prime-power factorization of n!. 29
0, 0, 1, 2, 4, 5, 7, 8, 11, 13, 15, 16, 19, 20, 22, 24, 28, 29, 32, 33, 36, 38, 40, 41, 45, 47, 49, 52, 55, 56, 59, 60, 65, 67, 69, 71, 75, 76, 78, 80, 84, 85, 88, 89, 92, 95, 97, 98, 103, 105, 108, 110, 113, 114, 118, 120, 124, 126, 128, 129, 133, 134, 136, 139 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Daniel Forgues, Table of n, a(n) for n = 0..100000

Mehdi Hassani, On the decomposition of n! into primes, arXiv:math/0606316 [math.NT], 2006-2007.

Keith Matthews, Computing the prime-power factorization of n!

FORMULA

a(n) = a(n-1) + A001222(n).

A027746(a(A000040(n))+1) = A000040(n). A082288(a(n)+1) = n.

A001221(n!) = omega(n!) = pi(n) = A000720(n).

a(n) = Sum_{i = 1..n} A001222(i). - Jonathan Vos Post, Feb 10 2010

a(n) = n log log n + B_2 * n + o(n), with B_2 = A083342. - Charles R Greathouse IV, Jan 11 2012

a(n) = A210241(n) - n for n > 0. - Reinhard Zumkeller, Mar 23 2012

G.f.: (1/(1 - x))*Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Mar 15 2017

EXAMPLE

For n=5, 5! = 120 = 2^3*3^1*5^1 so a(5) = 3+1+1 = 5. - N. J. A. Sloane, May 26 2018

MAPLE

with(numtheory):with(combinat):a:=proc(n) if n=0 then 0 else bigomega(numbperm(n)) fi end: seq(a(n), n=0..63); # Zerinvary Lajos, Apr 11 2008

MATHEMATICA

Array[Plus@@Last/@FactorInteger[ #! ] &, 5!, 0] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

f[n_] := If[n <= 1, 0, Total[FactorInteger[n]][[2]]]; Accumulate[Array[f, 100, 0]] (* T. D. Noe, Apr 11 2011 *)

Table[PrimeOmega[n!], {n, 0, 70}] (* Jean-Fran├žois Alcover, Jun 08 2013 *)

Join[{0}, Accumulate[PrimeOmega[Range[70]]]] (* Harvey P. Dale, Jul 23 2013 *)

PROG

(PARI) a(n)=bigomega(n!)

(PARI) k=0; vector(n, i, k+=bigomega(i))

(PARI) a(n) = sum(k=1, primepi(n), (n - sumdigits(n, prime(k))) / (prime(k)-1)); \\ Daniel Suteu, Apr 18 2018

(PARI) a(n) = my(res = 0); forprime(p = 2, n, cn = n; while(cn > 0, res += (cn \= p))); res \\ David A. Corneth, Apr 27 2018

(Haskell)

a022559 n = a022559_list !! n

a022559_list = scanl (+) 0 $ map a001222 [1..]

-- Renhard Zumkeller, Feb 16 2012

CROSSREFS

Cf. A001222, A013939, A046660, A144494, A115627, A238002.

Sequence in context: A111040 A191324 A147807 * A049781 A076697 A001606

Adjacent sequences:  A022556 A022557 A022558 * A022560 A022561 A022562

KEYWORD

nonn,nice

AUTHOR

Karen E. Wandel (kw29(AT)evansville.edu)

EXTENSIONS

Typo corrected by Daniel Forgues, Nov 16 2009

STATUS

approved

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Last modified August 16 00:25 EDT 2018. Contains 313782 sequences. (Running on oeis4.)