A007468 Sum of next n primes. (Formerly M1846)

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

1,1

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

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A359117 A294264 A367037 * A280156 A054137 A343414

Adjacent sequences: A007465 A007466 A007467 * A007469 A007470 A007471

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Zak Seidov, Sep 21 2002

Status

approved