A034387 - OEIS

A034387

Sum of primes <= n.

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, 381, 381, 381, 381, 381

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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

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

LINKS

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

Jens Askgaard, On the additive period length of the Sprague-Grundy function of certain Nim-like games, arXiv:1902.06299 [math.CO], 2019.

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), Apr 07 2001

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

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

a(n) = (n+1)*A000720(n) - A046992(n). - Ridouane Oudra, Sep 18 2021

a(n) = A007504(A000720(n)). - Ridouane Oudra, Feb 22 2022

a(n) = Sum_{p<=n, p prime} p. - Wesley Ivan Hurt, Dec 31 2023

MAPLE

a:= proc(n) option remember; `if`(n<1, 0,

a(n-1)+`if`(isprime(n), n, 0))

end:

seq(a(n), n=1..60); # Alois P. Heinz, Jun 29 2022

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

(PARI) a(n) = if(n <= 1, return(0)); my(r=sqrtint(n)); my(V=vector(r, k, n\k)); my(L=n\r-1); V=concat(V, vector(L, k, L-k+1)); my(T=vector(#V, k, V[k]*(V[k]+1)\2)); my(S=Map(matrix(#V, 2, x, y, if(y==1, V[x], T[x])))); forprime(p=2, r, my(sp=mapget(S, p-1), p2=p*p); for(k=1, #V, if(V[k] < p2, break); mapput(S, V[k], mapget(S, V[k]) - p*(mapget(S, V[k]\p) - sp)))); mapget(S, n)-1; \\ Daniel Suteu, Jun 29 2022

(Haskell)

a034387 n = a034387_list !! (n-1)

a034387_list = scanl1 (+) a061397_list

-- Reinhard Zumkeller, Mar 21 2014

(Python)

from sympy import isprime

from itertools import accumulate

def alist(n): return list(accumulate(k*isprime(k) for k in range(1, n+1)))

print(alist(57)) # Michael S. Branicky, Sep 18 2021

CROSSREFS

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

This is a lower bound on A287881.