A071149 -id:A071149 - cp's OEIS frontend

A071150 Primes p such that the sum of all odd primes <= p is also a prime.

3, 29, 53, 61, 251, 263, 293, 317, 359, 383, 503, 641, 647, 787, 821, 827, 911, 1097, 1163, 1249, 1583, 1759, 1783, 1861, 1907, 2017, 2287, 2297, 2593, 2819, 2837, 2861, 3041, 3079, 3181, 3461, 3541, 3557, 3643, 3779, 4259, 4409, 4457, 4597, 4691, 4729, 4789

Offset: 1

Labos Elemer, May 13 2002

			29 is a prime and 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 127 (also a prime), so 29 is a term. - _Jon E. Schoenfield_, Mar 29 2021

R. J. Mathar, Table of n, a(n) for n = 1..5908

Analogous to A013917.

Cf. A065091, A071148, A071149, A013916, A013917, A013918, A007504.

SoddP := proc(n)
    option remember;
    if n <= 2 then
        0;
    elif isprime(n) then
        procname(n-1)+n;
    else
        procname(n-1);
    fi ;
end proc:
isA071150 := proc(n)
    if isprime(n) and isprime(SoddP(n)) then
        true;
    else
        false;
    end if;
end proc:
n := 1 ;
for i from 3 by 2 do
    if isA071150(i) then
        printf("%d %d\n",n,i) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Feb 13 2015

Function[s, Select[Array[Take[s, #] &, Length@ s], PrimeQ@ Total@ # &][[All, -1]]]@ Prime@ Range[2, 640] (* Michael De Vlieger, Jul 18 2017 *)
Module[{nn=650,pr},pr=Prime[Range[2,nn]];Table[If[PrimeQ[Total[Take[ pr, n]]], pr[[n]],Nothing],{n,nn-1}]] (* Harvey P. Dale, May 12 2018 *)

from sympy import isprime, nextprime
def aupto(limit):
  p, s, alst = 3, 3, []
  while p <= limit:
    if isprime(s): alst.append(p)
    p = nextprime(p)
    s += p
  return alst
print(aupto(4789)) # Michael S. Branicky, Mar 29 2021

A007610 Sum of n consecutive primes starting at a(n) is prime (or 0 if impossible).

Original entry on oeis.org

2, 2, 5, 2, 5, 2, 17, 0, 3, 0, 5, 2, 29, 2, 3, 0, 3, 0, 11, 0, 7, 0, 7, 0, 5, 0, 7, 0, 13, 0, 13, 0, 7, 0, 5, 0, 5, 0, 13, 0, 7, 0, 7, 0, 7, 0, 7, 0, 11, 0, 17, 0, 3, 0, 3, 0, 97, 0, 29, 2, 3, 0, 13, 2, 3, 0, 19, 0, 19, 0, 3, 0, 5, 0, 3, 0, 23, 0, 7, 0, 11, 0, 53, 0, 31, 0, 89, 0, 53, 0, 19, 0, 11, 0, 3, 2

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

a(n) = 0 iff n is even and the sum of 2..P(n) is not prime. See A013916. - Robert G. Wilson v, Feb 16 2002

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. W. Trigg, Prime sums of consecutive primes, J. Rec. Math., 18 (No. 4, 1985-1986), 247-248.

T. D. Noe, Table of n, a(n) for n = 1..10000
C. W. Trigg, Prime sums of consecutive primes, J. Rec. Math., 18 (No. 4, 1985-1986), 247-248. (Annotated scanned copy)

Cf. A013916, A071149.

f[n_] := If[OddQ@ n, Block[{k = 1}, While[ !PrimeQ[Plus @@ Prime[Range[k, k + n - 1]]], k++]; Prime@ k], If[ PrimeQ[Plus @@ Prime@ Range@ n], 2, 0]]; Array[f, 96] (* Robert G. Wilson v, May 11 2015 *)

A376891 Numbers k such that the sum of the first k lesser of twin primes is a lesser of twin prime.

Original entry on oeis.org

1, 23, 143, 251, 281, 305, 341, 455, 605, 761, 1349, 1613, 2765, 2903, 2981, 3623, 3725, 3923, 4049, 4133, 4745, 5207, 5303, 5489, 5765, 6515, 6611, 7793, 7835, 8153, 8237, 10427, 10697, 11261, 11447, 11627, 11729, 12401, 12701, 13871, 14327, 15359, 15683

Offset: 1

Ilya Gutkovskiy, Oct 08 2024

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A001359, A013916, A071149, A086167, A089228, A376892.

K:= 1: count:= 1: s:= 3: k:= 1:
for p from 5 by 6 do
  if isprime(p) and isprime(p+2) then
    k:= k+1;
    s:= s+p;
    if s mod 6 = 5 and isprime(s) and isprime(s+2) then
      count:= count+1; K:= K,k;
      if count = 100 then break fi;
fi fi od:
K; # Robert Israel, Oct 08 2024

Position[Accumulate[Select[Partition[Prime[Range[200000]],2,1],#[[2]]-#[[1]]==2&][[;;,1]]],?(AllTrue[#+{0,2},PrimeQ]&)]//Quiet//Flatten (* _Harvey P. Dale, Jun 24 2025 *)

lista(nn) = my(v=select(p->isprime(p+2), primes(nn)), s = vector(#v)); s[1] = v[1]; for (i=2, #v, s[i] = s[i-1]+v[i]); Vec(select(x->(isprime(x) && isprime(x+2)), s, 1)); \\ Michel Marcus, Oct 10 2024

A376892 Numbers k such that the sum of the first k greater of twin primes is a greater of twin prime.

Original entry on oeis.org

1, 45, 105, 675, 987, 1431, 1593, 1677, 1785, 1875, 2037, 2541, 3039, 3045, 3051, 3183, 3267, 3531, 3699, 4113, 4239, 4377, 4443, 5643, 5673, 5709, 6027, 6543, 6615, 6771, 6891, 6915, 6999, 8043, 8109, 8313, 8607, 8739, 10197, 10569, 11103, 11139, 11361, 11787, 12045

Offset: 1

Ilya Gutkovskiy, Oct 08 2024

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A006512, A013916, A071149, A086168, A089228, A376891.

K:= 1: count:= 1: s:= 5: k:= 1:
for p from 7 by 6 do
  if isprime(p) and isprime(p-2) then
    k:= k+1;
    s:= s+p;
    if s mod 6 = 1 and isprime(s) and isprime(s-2) then
      count:= count+1; K:= K, k;
      if count = 100 then break fi;
fi fi od:
K; # Robert Israel, Nov 08 2024

lista(nn) = my(v=select(p->isprime(p-2), primes(nn)), s = vector(#v)); s[1] = v[1]; for (i=2, #v, s[i] = s[i-1]+v[i]); Vec(select(x->(isprime(x) && isprime(x-2)), s, 1)); \\ Michel Marcus, Oct 10 2024

A372041 Least prime p such that the sum of squares of the 2n + 1 consecutive primes starting with p is prime, or -1 if no such p exists.

Original entry on oeis.org

3, 3, 5, 3, 3, 5, -1, 5, 5, -1, 3, 7, -1, 3, 13, -1, 5, 5, -1, 7, 23, -1, 13, 5, -1, 7, 5, -1, 59, 29, 3, 3, 5, -1, 3, 5, -1, 13, 11, -1, 37, 23, -1, 43, 11, -1, 3, 5, -1, 11, 5, -1, 5, 19, -1, 5, 43, -1, 13, 29, -1, 7, 19, -1, 41, 47, -1, 13, 11, 3, 7, 5, -1, 29, 7, -1, 79, 13, 3, 3

Offset: 1

Michel Lagneau, Apr 17 2024

sign

a(n) = 2 never occurs, since the sum starting at 2 is always even and >= 4, so not prime.
a(n) = 3 iff n is in A370633 (and equivalently iff 2*n+1 is in A071149).
For n == 1 (mod 3), so 2*n+1 is a multiple of 3, a(n) = 3 or -1, since all primes >= 5 are congruent to 1 (mod 6) so the sum starting at 5 or more is a multiple of 3 and so not prime.

			a(6) = 5 because 5 is the smallest of 2*6+1 = 13 consecutive primes whose sum of squares = 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 31^2 + 37^2 + 41^2 + 43^2 + 47^2 = 10453 is prime.
a(7) = -1 because 7 == 1 (mod 3) so its only possibility is that the sum starts at 3, but 3^2 + ... + 53^2 = 13271 is not prime.

Cf. A024450, A089793, A318351, A340771, A370633 (indices of 3's).

a(n) = if ((n % 3) == 1, my(vp = primes(2*n+2)); if (isprime(sum(k=2, #vp, vp[k]^2)), return (3), return(-1));); my(vp = primes(2*n+2)); while(! isprime(sum(k=2, #vp, vp[k]^2)), vp = concat(setminus(vp, Set(vp[1])), nextprime(vp[2*n+2]+1))); vp[2]; \\ Michel Marcus, May 16 2024

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A071150 Primes p such that the sum of all odd primes <= p is also a prime.

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A007610 Sum of n consecutive primes starting at a(n) is prime (or 0 if impossible).

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A376891 Numbers k such that the sum of the first k lesser of twin primes is a lesser of twin prime.

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A376892 Numbers k such that the sum of the first k greater of twin primes is a greater of twin prime.

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A372041 Least prime p such that the sum of squares of the 2n + 1 consecutive primes starting with p is prime, or -1 if no such p exists.

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PARI