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A013939 Partial sums of sequence A001221 (number of distinct primes dividing n). 43
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
a(n) = Sum_{k>=1} k * A346617(n,k). - Alois P. Heinz, Aug 19 2021
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. A022559.
Cf. A077761.
Cf. A346617.
Sequence in context: A008320 A004439 A050126 * A343109 A209921 A268377
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Henry Bottomley, Jul 03 2001
STATUS
approved

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Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)