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 A013939 Partial sums of sequence A001221 (number of distinct primes dividing n). 33
 0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 34, 36, 37, 39, 40, 43, 44, 45, 47, 49, 51, 53, 54, 56, 58, 60, 61, 64, 65, 67, 69, 71, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 93, 96, 97, 99, 101, 102, 104, 107, 108, 110, 112 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n) = A093614(n) - A048865(n); see also A006218. A027748(a(A000040(n))+1) = A000040(n), A082287(a(n)+1) = n. LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 Eric Weisstein's World of Mathematics, Distinct Prime Factors FORMULA a(n) = Sum_{k <= n} omega(k). a(n) = Sum_{k = 1..n} floor( n/prime(k) ). a(n) = a(n-1) + A001221(n). a(n) = Sum_{k=1..n} pi(floor(n/k)). - Vladeta Jovovic, Jun 18 2006 a(n) = n log log n + O(n). - Charles R Greathouse IV, Jan 11 2012 a(n) = n*(log log n + B) + o(n), where B = 0.261497... is the Mertens constant A077761. - Arkadiusz Wesolowski, Oct 18 2013 G.f.: (1/(1 - x))*Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 02 2017 a(n) = Sum_{k=1..floor(sqrt(n))} k * (pi(floor(n/k)) - pi(floor(n/(k+1)))) + Sum_{p prime <= floor(n/(1+floor(sqrt(n))))} floor(n/p). - Daniel Suteu, Nov 24 2018 MAPLE A013939 := proc(n) option remember;  `if`(n = 1, 0, a(n) + iquo(n+1, ithprime(n+1))) end: seq(A013939(i), i = 1..69);  # Peter Luschny, Jul 16 2011 MATHEMATICA a[n_] := Sum[Floor[n/Prime[k]], {k, 1, n}]; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Jun 11 2012, from 2nd formula *) Accumulate[PrimeNu[Range[120]] (* Harvey P. Dale, Jun 05 2015 *) PROG (PARI) t=0; vector(99, n, t+=omega(n)) \\ Charles R Greathouse IV, Jan 11 2012 (PARI) a(n)=my(s); forprime(p=2, n, s+=n\p); s \\ Charles R Greathouse IV, Jan 11 2012 (PARI) a(n) = sum(k=1, sqrtint(n), k * (primepi(n\k) - primepi(n\(k+1)))) + sum(k=1, n\(sqrtint(n)+1), if(isprime(k), n\k, 0)); \\ Daniel Suteu, Nov 24 2018 (Haskell) a013939 n = a013939_list !! (n-1) a013939_list = scanl1 (+) \$ map a001221 [1..] -- Reinhard Zumkeller, Feb 16 2012 (Python) from sympy.ntheory import primefactors print([sum(len(primefactors(k)) for k in range(1, n+1)) for n in range(1, 121)]) # Indranil Ghosh, Mar 19 2017 (MAGMA) [(&+[Floor(n/NthPrime(k)): k in [1..n]]): n in [1..70]]; // G. C. Greubel, Nov 24 2018 (Sage) [sum(floor(n/nth_prime(k)) for k in (1..n)) for n in (1..70)] # G. C. Greubel, Nov 24 2018 CROSSREFS Cf. A005187, A006218, A011371, A013936. Cf. A022559. Cf. A077761. Sequence in context: A008320 A004439 A050126 * A209921 A268377 A201010 Adjacent sequences:  A013936 A013937 A013938 * A013940 A013941 A013942 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Henry Bottomley, Jul 03 2001 STATUS approved

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Last modified November 30 22:47 EST 2020. Contains 338831 sequences. (Running on oeis4.)