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 A022559 Sum of exponents in prime-power factorization of n!. 53
 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! Daniel Suteu, Perl program 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 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 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 # Alternative: ListTools:-PartialSums(map(numtheory:-bigomega, [\$0..200])); # Robert Israel, Dec 21 2018 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-François 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..] -- Reinhard 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 October 15 22:25 EDT 2019. Contains 328038 sequences. (Running on oeis4.)