A086167 -id:A086167 - cp's OEIS frontend

A270563 Integers k such that A086167(k) and A086168(k) are both prime.

1, 15, 45, 105, 135, 231, 807, 1215, 1329, 1395, 1593, 1911, 2301, 2331, 2493, 3045, 3267, 3417, 3495, 3897, 4029, 4059, 4359, 4377, 4635, 4665, 4731, 5265, 6135, 6315, 6429, 6489, 6795, 6915, 6999, 7329, 7515, 7965, 8469, 8979, 9183, 9441, 10755, 11193, 12039

Offset: 1

Altug Alkan, Mar 19 2016

nonn

A013916 lists numbers n such that the sum of the first n primes is prime. With similar motivation, twin prime pairs generate prime pairs in this sequence. Note that 2*n also gives the difference between members of prime pair that is generated by sum of first n twin prime pairs.

First differences of this sequence are 14, 30, 60, 30, 96, 576, ...

			15 is a term since A086167(15) = 1297 and A086168(15) = 1297 + 15*2 = 1327. 1297 and 1327 are both prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A086167, A086168.

seq = {}; s1 = s2 = 0; c = n = 0; p = prv = 2; While[c < 45, p = NextPrime[p]; If[p == prv + 2, n++; s1 += prv; s2 += p; If[PrimeQ[s1] && PrimeQ[s2], c++; AppendTo[seq, n]]]; prv = p]; seq (* Amiram Eldar, Jan 03 2020 *)

t(n, p=3) = {while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
s1(n) = sum(k=1, n, t(k));
s2(n) = sum(k=1, n, t(k)+2);
for(n=1, 1e3, if(ispseudoprime(s1(n)) && ispseudoprime(s2(n)), print1(n, ", ")));

More terms from Amiram Eldar, Jan 03 2020

A086168 a(n) = sum of the first n upper twin primes.

Original entry on oeis.org

5, 12, 25, 44, 75, 118, 179, 252, 355, 464, 603, 754, 935, 1128, 1327, 1556, 1797, 2068, 2351, 2664, 3013, 3434, 3867, 4330, 4853, 5424, 6025, 6644, 7287, 7948, 8759, 9582, 10411, 11270, 12153, 13174, 14207, 15258, 16321, 17414, 18567

Offset: 1

Cino Hilliard, Aug 25 2003

nonn

			For n = 4 we have twin prime pairs (3,5) (5,7) (11,13) (17,19) and 5 + 7 + 13 + 19 = 44.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006512, A086167.

lst={};s=0;Do[p=Prime[n];If[PrimeQ[p-2], s+=p;AppendTo[lst, s]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 30 2008 *)
Accumulate[Transpose[Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]==2&]][[2]]]  (* Harvey P. Dale, Feb 08 2011 *)

addnexttwin(n)= { s=0; for(x=1,n, if(prime(x+1)-prime(x)==2,s=s+prime(x+1); print1(s",")) ) }

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

A360226 a(n) = sum of the first n primes whose distance to next prime is 4.

Original entry on oeis.org

7, 20, 39, 76, 119, 186, 265, 362, 465, 574, 701, 864, 1057, 1280, 1509, 1786, 2093, 2406, 2755, 3134, 3531, 3970, 4427, 4890, 5377, 5876, 6489, 7132, 7805, 8544, 9301, 10070, 10893, 11746, 12605, 13482, 14365, 15272, 16209, 17176, 18185, 19272, 20365, 21578, 22857, 24154, 25457, 26880, 28309, 29756

Offset: 1

Artur Jasinski, Feb 01 2023

Sidney Cadot, Table of n, a(n) for n = 1..1000

Cf. A007504, A029710, A086167, A086168, A172112.

ii = {}; sum = 0; Do[If[Prime[n + 1] - Prime[n] == 4, sum = sum + Prime[n]; AppendTo[ii, sum]], {n, 1, 250}]; ii

a(n) = Sum_{k=1..n} A029710(k).

a(n) = A172112(n+1) - 3.

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A270563 Integers k such that A086167(k) and A086168(k) are both prime.

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A086168 a(n) = sum of the first n upper twin primes.

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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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A360226 a(n) = sum of the first n primes whose distance to next prime is 4.

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