A007468 Sum of next n primes.

2, 8, 31, 88, 199, 384, 659, 1056, 1601, 2310, 3185, 4364, 5693, 7360, 9287, 11494, 14189, 17258, 20517, 24526, 28967, 33736, 38917, 45230, 51797, 59180, 66831, 75582, 84463, 95290, 106255, 117424, 129945, 143334, 158167, 173828, 190013, 207936, 225707, 245724

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

If we arrange the prime numbers into a triangle, with 2 at the top, 3 and 5 in the second row, 7, 11 and 13 in the third row, and so on and so forth, this sequence gives the row sums. - Alonso del Arte, Nov 08 2011

In the first 20000 terms, the only perfect square > 1 is 207936 (n=38). Is it the only one? Is there some proof/conjecture? - Carlos Eduardo Olivieri, Mar 09 2015

Examples

			a(1)=2 because "sum of next 1 prime" is 2;
a(2)=8 because sum of next 2 primes is 3+5=8;
a(3)=31 because sum of next 3 primes is 7+11+13=31, etc.
		

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A078721 and A011756 for the starting and ending prime of each sum.

Programs

Mathematica

a[n_] := Sum[Prime[i], {i, 1+n(n-1)/2, n+n(n-1)/2}]; Table[a[n], {n,100}]
(* Second program: *)
With[{nn=40},Total/@TakeList[Prime[Range[(nn(nn+1))/2]],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jan 15 2020 *)

Python

from sympy import nextprime
def aupton(terms):
  alst, p = [], 2
  for n in range(1, terms+1):
    s = 0
    for i in range(n):
      s += p
      p = nextprime(p)
    alst.append(s)
  return alst
print(aupton(40)) # Michael S. Branicky, Feb 08 2021

Formula

a(n) = prime(1 + n(n-1)/2) + ... + prime(n + n(n-1)/2), where prime(i) is i-th prime.

Extensions

More terms from Zak Seidov, Sep 21 2002