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A034387 Sum of primes <= n. 32
0, 2, 5, 5, 10, 10, 17, 17, 17, 17, 28, 28, 41, 41, 41, 41, 58, 58, 77, 77, 77, 77, 100, 100, 100, 100, 100, 100, 129, 129, 160, 160, 160, 160, 160, 160, 197, 197, 197, 197, 238, 238, 281, 281, 281, 281, 328, 328, 328, 328, 328, 328 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also sum of all prime-factors in n!.

For large n, these numbers can be closely approximated by the number of primes < n^2. For example, the sum of primes < 10^10 = 2220822432581729238. The number of primes < (10^10)^2 or 10^20 = 2220819602560918840. This has a relative error of 0.0000012743... - Cino Hilliard, Jun 08 2008

Equals row sums of triangle A143537. - Gary W. Adamson, Aug 23 2008

a(n) = A158662(n) - 1. a(p) - a(p-1) = p, for p = primes (A000040), a(c) - a(c-1) = 0, for c = composite numbers (A002808). - Jaroslav Krizek, Mar 23 2009

Partial sums of A061397. - Reinhard Zumkeller, Mar 21 2014

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Cino Hilliard, Sum of primes

FORMULA

From the prime number theorem a(n) has the asymptotic expression: a(n) ~ n^2 / (2 log n) - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

a(n) = n^2/(2 log n) + O(n^2 log log n/log^2 n). - Vladimir Shevelev and Charles R Greathouse IV, May 29 2014

Conjecture: G.f.: Sum_{i>0} Sum_{j>=i} Sum_{k>=j|i-j+k is prime} x^k. - Benedict W. J. Irwin, Mar 31 2017

MATHEMATICA

s=0; Table[s=s+n*Boole[PrimeQ[n]], {n, 100}] (* Zak Seidov, Apr 11 2011 *)

Accumulate[Table[If[PrimeQ[n], n, 0], {n, 60}]] (* Harvey P. Dale, Jul 25 2016 *)

PROG

(PARI) a(n)=sum(i=1, primepi(n), prime(i)) \\ Michael B. Porter, Sep 22 2009

(PARI) a=0; for(k=1, 100, print1(a=a+k*isprime(k), ", ")) \\ Zak Seidov, Apr 11 2011

(Haskell)

a034387 n = a034387_list !! (n-1)

a034387_list = scanl1 (+) a061397_list

-- Reinhard Zumkeller, Mar 21 2014

CROSSREFS

Cf. A007504, A158662, A073837, A066779, A034386, A000720.

Sequence in context: A173567 A265129 A212624 * A081240 A184443 A238655

Adjacent sequences:  A034384 A034385 A034386 * A034388 A034389 A034390

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified May 30 08:46 EDT 2017. Contains 287302 sequences.