A007504 - OEIS (original) (raw)

0, 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160, 197, 238, 281, 328, 381, 440, 501, 568, 639, 712, 791, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 1988, 2127, 2276, 2427, 2584, 2747, 2914, 3087, 3266, 3447, 3638, 3831, 4028, 4227, 4438, 4661, 4888

COMMENTS

This is true. Proof: By definition we have A034960(n) = Sum_{k = (a(n-1)+1)..a(n)} (2*k-1). Since Sum_{k = 1..n} (2*k-1) = n^2, it follows A034960(n) = a(n)^2 - a(n-1)^2, for n > 1. - Hieronymus Fischer, Sep 27 2012 [formulas above adjusted to changed offset of A034960 - Hieronymus Fischer, Oct 14 2012]

Ramanujan noticed the apparent identity between the prime parts partition numbers A000607 and the expansion of Sum_{k >= 0} x^a(k)/((1-x)...(1-x^k)), cf. A046676. See A192541 for the difference between the two. - M. F. Hasler, Mar 05 2014

a(n) is the smallest number that can be partitioned into n distinct primes. - Alonso del Arte, May 30 2017

For a(n) < m < a(n+1), n > 0, at least one m is a perfect square.

Proof: For n = 1, 2, ..., 6, the proposition is clear. For n > 6, a(n) < ((prime(n) - 1)/2)^2, set (k - 1)^2 <= a(n) < k^2 < ((prime(n) + 1)/2)^2, then k^2 < (k - 1)^2 + prime(n) <= a(n) + prime(n) = a(n+1), so m = k^2 is this perfect square. - Jinyuan Wang, Oct 04 2018

For n >= 5 we have a(n) < ((prime(n)+1)/2)^2. This can be shown by noting that ((prime(n)+1)/2)^2 - ((prime(n-1)+1)/2)^2 - prime(n) = (prime(n)+prime(n-1))*(prime(n)-prime(n-1)-2)/4 >= 0. - Jianing Song, Nov 13 2022

Washington gives an oscillation formula for |a(n) - pi(n^2)|, see links. - Charles R Greathouse IV, Dec 07 2022

Conjecture: Each integer n >= 5340 can be written as a(i) + a(j) + 2*a(k) for some i,j,k > 0. (This has been verified for n <= 10^5.) - Zhi-Wei Sun, Oct 09 2025

Conjecture: Any integer n > 16866 can be written as p + a(k) with p prime and k positive. Also, each integer n > 13392 can be written as q + a(k) with q practical and k positive. (This has been verified for n <= 2*10^7.) - Zhi-Wei Sun, Oct 15 2025

REFERENCES

E. Bach and J. Shallit, §2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms, MIT Press, Cambridge, MA, 1996.

H. L. Nelson, "Prime Sums", J. Rec. Math., 14 (1981), 205-206.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Eric Weisstein's World of Mathematics, Prime Sums

FORMULA

a(n) ~ n^2 * log(n) / 2. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 24 2001 (see Bach & Shallit 1996)

For n >= 3, a(n) >= (n-1)^2 * (log(n-1) - 1/2)/2 and a(n) <= n*(n+1)*(log(n) + log(log(n))+ 1)/2. Thus a(n) = n^2 * log(n) / 2 + O(n^2*log(log(n))). It is more precise than in Fares's comment. - Vladimir Shevelev, Aug 01 2013

a(n) = (n^2/2)*(log n + log log n - 3/2 + (log log n - 3)/log n + (2 (log log n)^2 - 14 log log n + 27)/(4 log^2 n) + O((log log n/log n)^3)) [Sinha]. - Charles R Greathouse IV, Jun 11 2015

MAPLE

s1:=[2]; for n from 2 to 1000 do s1:=[op(s1), s1[n-1]+ithprime(n)]; od: s1;

add(ithprime(i), i=1..n) ;

MATHEMATICA

Accumulate[Prime[Range[100]]] (* Zak Seidov, Apr 10 2011 *)

primeRunSum = 0; Table[primeRunSum = primeRunSum + Prime[k], {k, 100}] (* Zak Seidov, Apr 16 2011 *)

PROG

(PARI) a(n) = vecsum(primes(n)); \\ Michel Marcus, Feb 06 2021

(Magma) [0] cat [&+[ NthPrime(k): k in [1..n]]: n in [1..50]]; // Bruno Berselli, Apr 11 2011 (adapted by Vincenzo Librandi, Nov 27 2015 after Hasler's change on Mar 05 2014)

(Haskell)

a007504 n = a007504_list !! n

a007504_list = scanl (+) 0 a000040_list

(GAP) P:=Filtered([1..250], IsPrime);;

a:=Concatenation([0], List([1..Length(P)], i->Sum([1..i], k->P[k]))); # Muniru A Asiru, Oct 07 2018

(Python)

from itertools import accumulate, count, islice

from sympy import prime

def A007504_gen(): return accumulate(prime(n) if n > 0 else 0 for n in count(0))

CROSSREFS

Cf. A000041, A034386, A111287, A013916, A013918 (primes), A045345, A050247, A050248, A068873, A073619, A034387, A014148, A014150, A178138, A254784, A254858, A387790.

See A122989 for the value of Sum_{n >= 1} 1/a(n).