A013916 -id:A013916 - cp's OEIS frontend

A121248 Numbers k such that the sum of the first 2^k primes is a prime.

0, 1, 2, 6, 9, 10, 32, 36, 55

Offset: 1

Alexander Adamchuk, Aug 22 2006

Corresponding primes in the sums of the first 2^n primes or primes in A099825[n] are given in A113617[n] = {2,5,17,8893,868151,3875933,219554912086470964379,...}.

Cf. A099825, A013916, A013918, A113617.

Do[f=Sum[Prime[k],{k,1,2^n}]; If[PrimeQ[f],Print[{n,f}]],{n,0,32}]

A099825(a(n)) = A113617(n). - Amiram Eldar, Jul 01 2024

Edited by Robert G. Wilson v, Aug 26 2006

a(7)-a(9) from Amiram Eldar, Jul 01 2024

A321439 Numbers k such that if j is the sum of the first prime(k) primes then the sum of the first j primes is prime.

Original entry on oeis.org

8, 21, 27, 37, 59, 65, 66, 82, 86, 99, 105, 111, 126, 143, 147, 155, 156, 165, 177, 181, 187, 194, 195, 200, 230, 231, 242, 262, 284, 374, 430, 449, 460, 477, 502, 512, 539, 540, 541, 622, 634, 657, 707, 731, 735, 739, 745, 766, 767, 781, 784, 785, 791, 801

Offset: 1

David James Sycamore, Nov 09 2018

nonn

Numbers k such that A007504(A007504(prime(k))) is prime. Terms can be even or odd since A007504(A007504(prime(k))) is odd for any k.

			8 is a term because prime(8) = 19, A007504(19) = 568, and A007504(568) = 1086557, which is prime.
2 is not a term since prime(2) = 3, A007504(3) = 10 and A007504(10) = 129, which is not prime.

Ray Chandler, Table of n, a(n) for n = 1..2500

Cf. A007504, A013916, A321342, A321343.

N:=100:
for n from 1 to N do
X:=add(ithprime(k),k=1..ithprime(n));
Y:=add(ithprime(r),r=1..X);
if isprime(Y)then print(n);
end if:
end do:

primeSum[n_] := Sum[Prime[i], {i, n}]; Select[Range[200], PrimeQ[ primeSum[primeSum[Prime[#]]]] &] (* Amiram Eldar, Nov 09 2018 *)

sumprimes(n)={my(p=0, s=0); for(i=1, n, p=nextprime(1+p); s+=p); s}
ok(k)={isprime(sumprimes(sumprimes(prime(k))))}
for(n=1, 100, if(ok(n),print1(n, ", "))) \\ Andrew Howroyd, Nov 11 2018

use ntheory qw(:all);
for (my ($i, $k) = (1, 1); ; ++$k) {
    if (is_prime sum_primes nth_prime sum_primes nth_prime nth_prime $k) {
        print "a($i) = $k\n"; ++$i;
    }
} # Daniel Suteu, Nov 11 2018

a(30)-a(54) from Daniel Suteu, Nov 11 2018

A321578 a(n) is the maximum value of k such that A007504(k) <= prime(n).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

David James Sycamore, Nov 12 2018

nonn

Let A be A007504. The number of distinct values of k such that a(k)=r is the number of primes p in the interval A(r) <= p < A(r+1); namely: 2,2,2,3,3,4,5,4,6,6,... (see A323701). Let b(n) be the smallest r such that a(r)=n, namely: 1,3,5,7,10,13,17,22,26,... For given n, if k is the index of the smallest prime >= A(n), then b(n)=k. (The equality applies when n is a term of A013916.)

			a(1)=1 since prime(1)=2 and 1 is max k such that A007504(k) <= 2.
a(5)=3 since prime(5)=11 and 3 is max k such that A007504(k) <= 11.
n=4 (in A013916). A(4)=17=prime(7), so b(4)=7.
n=7 (not in A013916). A(7)=58 < 59=prime(17), so b(7)=17.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A007504, A013916, A323701, A061568.

nn = 75; MapIndexed[Set[s[First[#2]], #1] &, Accumulate[Prime@ Range[nn]]]; k = 1; Table[p = Prime[n]; While[s[k] <= p, k++]; k - 1, {n, nn}] (* Michael De Vlieger, Jun 29 2025 *)

a(n) = my(k=0, p=0, s=0); while(s <= prime(n), k++; p=nextprime(p+1); s+=p); k-1; \\ Michel Marcus, Feb 19 2019

use ntheory ':all'; sub a { my $p = nth_prime($[0]); my($s, $q) = (0, 2); while ($s <= $p) { $s += $q; $q = next_prime($q) }; prime_count($q-1)-1 }; print join(", ", map { a($) } 1..100), "\n"; # Daniel Suteu, Jan 26 2019

A364796 Numbers k such that the sum of the first k prime powers (not including 1) is a prime power.

Original entry on oeis.org

1, 2, 3, 6, 8, 13, 18, 20, 22, 37, 41, 43, 46, 62, 87, 89, 95, 99, 111, 115, 118, 124, 130, 146, 150, 160, 164, 168, 180, 192, 201, 205, 211, 221, 263, 283, 287, 315, 339, 352, 356, 364, 396, 398, 408, 418, 434, 442, 450, 476, 508, 512, 526, 534, 536, 548, 556, 582

Offset: 1

Ilya Gutkovskiy, Aug 08 2023

nonn

			8 is a term because the sum of the first 8 prime powers 2 + 3 + 4 + 5 + 7 + 8 + 9 + 11 = 49 is a prime power.

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

Cf. A013916, A013919, A013930, A246655, A364797.

Position[Accumulate[Select[Range[4000], PrimePowerQ]], _?PrimePowerQ, Heads -> False] // Flatten

list(lim) = {my(k = 0, s = 0); for(p = 1, lim, if(isprimepower(p), k++; s += p; if(isprimepower(s), print1(k, ", "))));} \\ Amiram Eldar, Jun 20 2025

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

A136443 Numbers m such that A102863(m) = 1.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Giovanni Teofilatto, Apr 03 2008

nonn

Except first term, numbers n such that the sum of the first n primes is composite: Essentially the complement of A013916.

Cf. A013916, A102863.

Join[{1},Flatten[Position[Accumulate[Prime[Range[80]]],?CompositeQ]]] (* _Harvey P. Dale, Apr 23 2015 *)

A155973 Smallest x such that prime(2n)x^(2n-1) + prime(2n-1)x^(2n-2) + prime(2n-2)x^(2n-3) +...+ prime(2)x^1 + 2x^0 evaluates to an odd prime.

Original entry on oeis.org

1, 1, 1, 11, 23, 1, 1, 75, 29, 27, 159, 27, 107, 179, 63, 93, 675, 593, 11, 1299, 153, 153, 197, 35, 31, 227, 297, 439, 33, 1, 133, 1, 3, 1071, 173, 153, 299, 5, 1443, 1275, 611, 1809, 941, 669, 537, 51, 151, 1, 131, 1, 1, 343, 199, 1, 279, 3, 1, 439, 597, 453, 1, 1, 1187, 391

Offset: 1

Cino Hilliard, Jan 31 2009

nonn

Conjecture: The number of 1's in this sequence is infinite.

a(n) = 1 if and only if 2n is in A013916.

			n=1: 3x + 2, prime for x = 1, so a(1) = 1.
n=2: 7x^3 + 5x^2 + 3x + 2, prime for x = 1, so a(2) = 1.
n=3: 13x^5 + 11x^4 + 7x^3 + 5x^2 + 3x + 2, prime for x = 1, so a(3) = 1.
n=4: 19x^7 + 17x^6 + 13x^5 + 11x^4 + 7x^3 + 5x^2 + 3x + 2, prime for x = 11, so a(4) = 11.

Charles R Greathouse IV, Table of n, a(n) for n = 1..200

primenomial(n) = { ct=0; sr=0; p=0; d=0; d1=0; forstep(m=1,n,2, for(x=0,n,y=2; for(j=2,m+1, p = prime(j); y+=x^(j-1)*p; );
if(y>2&&ispseudoprime(y),ct+=1; print1(x",");break ); )) }

a(n)=my(P=Polrev(primes(2*n)),k=1);while(!ispseudoprime(subst(P, 'x, k)), k+=2); k \\ Charles R Greathouse IV, Jan 15 2013

a(39)-a(64) from Charles R Greathouse IV, Jan 17 2013

A228490 Numbers k for which prime(1) + ... + prime(k) = 2*prime(m) for some m.

Original entry on oeis.org

3, 7, 55, 59, 65, 75, 93, 133, 137, 141, 249, 277, 313, 365, 375, 387, 391, 435, 471, 499, 563, 573, 597, 605, 619, 645, 675, 719, 787, 797, 799, 815, 825, 845, 867, 879, 919, 937, 957, 971, 1011, 1013, 1145, 1217, 1225, 1243, 1251, 1271, 1283, 1311, 1373

Offset: 1

Clark Kimberling, Oct 01 2013

nonn

			a(1) = 3 because 2 + 3 + 5 = 2*prime(3) and p(3) = 5 is the least such prime summand.

Cf. A013916, A228491.

z = 2800; f[n_] := Sum[Prime[k], {k, 1, n}]; p[n_] := If[PrimeQ[f[n]/2], 1, 0]; t = Table[p[n], {n, 1, z}]; Flatten[Position[t, 1]]

A228491 Numbers k for which sum(first k primes) = 3*prime(m) for some m.

Original entry on oeis.org

10, 16, 18, 28, 32, 34, 36, 44, 46, 54, 82, 136, 138, 246, 250, 276, 286, 362, 370, 378, 390, 554, 570, 586, 588, 668, 678, 684, 688, 690, 726, 766, 770, 826, 856, 860, 878, 880, 888, 924, 928, 932, 956, 962, 1048, 1160, 1174, 1210, 1264, 1286, 1292, 1506

Offset: 1

Clark Kimberling, Oct 01 2013

nonn

			a(1) = 10 because 2 + 3 + 5 + ... + 29 = 3*prime(14) and p(10) = 29 is the least such prime summand.

Cf. A013916, A228490.

z = 2800; f[n_] := Sum[Prime[k], {k, 1, n}]; p[n_] := If[PrimeQ[f[n]/3], 1, 0]; t = Table[p[n], {n, 1, z}]; Flatten[Position[t, 1]]

cp's OEIS Frontend

A121248 Numbers k such that the sum of the first 2^k primes is a prime.

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A321439 Numbers k such that if j is the sum of the first prime(k) primes then the sum of the first j primes is prime.

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A321578 a(n) is the maximum value of k such that A007504(k) <= prime(n).

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A364796 Numbers k such that the sum of the first k prime powers (not including 1) is a prime power.

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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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A136443 Numbers m such that A102863(m) = 1.

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A155973 Smallest x such that prime(2n)x^(2n-1) + prime(2n-1)x^(2n-2) + prime(2n-2)x^(2n-3) +...+ prime(2)x^1 + 2x^0 evaluates to an odd prime.

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