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A022559
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Sum of exponents in prime-power factorization of n!.
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99
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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
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1/(1 - x))*Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Mar 15 2017
a(n) = Sum_{k=1..floor(sqrt(n))} k * (A025528(floor(n/k)) - A025528(floor(n/(k+1)))) + Sum_{k=1..floor(n/(floor(sqrt(n))+1))} floor(n/k) * A069513(k). - Daniel Suteu, Dec 21 2018
a(n) = Sum_{prime p<=n} Sum_{k=1..floor(log_p(n))} floor(n/p^k). - Ridouane Oudra, Nov 04 2022
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EXAMPLE
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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
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MAPLE
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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
# Alternative:
ListTools:-PartialSums(map(numtheory:-bigomega, [$0..200])); # Robert Israel, Dec 21 2018
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MATHEMATICA
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f[n_] := If[n <= 1, 0, Total[FactorInteger[n]][[2]]]; Accumulate[Array[f, 100, 0]] (* T. D. Noe, Apr 11 2011 *)
Join[{0}, Accumulate[PrimeOmega[Range[70]]]] (* Harvey P. Dale, Jul 23 2013 *)
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PROG
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(PARI) a(n)=bigomega(n!)
(PARI) first(n)={my(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..]
(Python)
from sympy import factorint as pf
def aupton(nn):
alst = [0]
for n in range(1, nn+1): alst.append(alst[-1] + sum(pf(n).values()))
return alst
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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Karen E. Wandel (kw29(AT)evansville.edu)
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EXTENSIONS
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STATUS
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approved
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