A013916 -id:A013916 - cp's OEIS frontend

A235431 The smallest positive number that must be added to or subtracted from the sum of the first n primes in order to get a prime.

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 4, 1, 2, 5, 2, 1, 4, 1, 4, 1, 2, 3, 4, 5, 2, 3, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 10, 1, 4, 11, 2, 1, 6

Offset: 1

R. J. Cano, Jan 17 2014

The primes in A013918 would have associated a(n)=0 if not for the qualifier "positive" in the definition.
The sum of the first n primes appears to be close to a prime. For illustration, the maximum for a(n) among the first 5 million terms is a(808500) = 218.
See A013916 for the above mentioned indices, numbers n such that the sum of the first n primes is prime. - M. F. Hasler, Jan 20 2014

			The sum of the first 9 primes is 100, and by adding 1 we get 101. Since 101 is a prime, a(9) = 1.
The sum of the first 10 primes is 129, since 129 - 2 = prime(31) = 127 or 129 + 2 = prime(32) = 131, a(10) = 2.
The sum of the first 129 primes minus 1 is a prime, this is 42468 - 1 = prime(4443), so a(129) = 1.

R. J. Cano, Table of n, a(n) for n = 1..10000

Cf. A007504, A101301.

a(n)=my(u=sum(j=1,n,prime(j)),k=1);while(!(isprime(u+k)||isprime(u-k)),k++);k

Algorithm:
Let S be the sum of the first n primes;
initially, let k=1;
increment k while neither S-k nor S+k is prime;
return a(n)=k.
a(n) = min(A013632(A007504(n)), A049711(A007504(n))). - M. F. Hasler, Jan 20 2014

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

Original entry on oeis.org

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

A277123 Numbers k such that 1 + Sum_{j=1..k} prime(j)^2 is prime.

Original entry on oeis.org

1, 11, 19, 29, 37, 73, 97, 155, 163, 175, 191, 257, 295, 313, 325, 341, 365, 389, 391, 409, 415, 461, 491, 497, 515, 599, 697, 715, 757, 761, 767, 775, 785, 793, 857, 875, 895, 899, 905, 919, 1099, 1109, 1117, 1139, 1151, 1163, 1225, 1271, 1279, 1295, 1309

Offset: 1

Alex Ratushnyak, Sep 30 2016

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A013916, A089228, A131694, A159260.

Position[Accumulate[Prime[Range[2000]]^2]+1,?PrimeQ]//Flatten (* _Harvey P. Dale, Sep 07 2019 *)

lista(nn) = for(n=1, nn, if(isprime(1+sum(i=1, n, prime(i)^2)), print1(n, ", "))); \\ Altug Alkan, Oct 01 2016

import sympy
sum = p = 1
for n in range(1,3001):
  while not sympy.isprime(p):  p+=1    # find the n'th prime
  sum += p*p
  p+=1
  if sympy.isprime(sum):  print(n, end=', ')

A280536 Numbers n such that the sum of the first n primes reduced by some power of 2 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 11, 12, 14, 15, 19, 21, 27, 35, 55, 59, 60, 64, 65, 75, 81, 83, 93, 95, 96, 100, 102, 108, 109, 114, 122, 124, 130, 132, 133, 135, 137, 141, 146, 152, 155, 158, 162, 165, 171, 178, 183, 192, 193, 198, 204, 206, 208, 214, 216, 223, 227, 243, 249, 255, 257, 263, 277, 279, 296

Offset: 1

Robert G. Wilson v, Jan 04 2017

nonn

A013916 except for the first term, 1, is a proper subset.
The odd terms are: 3, 5, 7, 11, 15, 19, 21, 27, 35, 55, 59, 65, 75, 81, 83, ..., .
The 10^k-th term: 2, 15, 469, 7980, 110374, 1359497, 16214466, 187663922, ..., .

			11 is in the sequence since the sum of the first 11 primes is A007504(11) = 160 and 160 is divisible by 2^5 which gives 5, a prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A007504, A013916.

fQ[n_] := Block[{s = Sum[ Prime@ k, {k, n}]}, PrimeQ[s/2^IntegerExponent[s, 2]]]; Select[Range@300, fQ]

A282246 Primes p such that the sum of all primes <= p has no prime divisor > p.

Original entry on oeis.org

2, 5, 11, 19, 23, 31, 41, 47, 59, 71, 83, 97, 101, 103, 109, 113, 127, 137, 157, 163, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 241, 263, 269, 271, 317, 337, 349, 353, 367, 389, 401, 409, 433, 439, 449, 457, 461, 463, 467, 491, 521, 563, 571, 607, 613, 617, 631, 641, 653, 661, 701, 709, 719, 739, 757, 797

Offset: 1

Emmanuel Vantieghem, Feb 09 2017

nonn

Number of terms < 10^k: 2, 12, 79, 523, 4124, 32678, 267850, etc. Compare these to A006880. - Robert G. Wilson v, Feb 09 2017

Primes p such that A006530(A007504(i)) <= p, where i is the index of p in A000040. - Felix Fröhlich, Feb 12 2017

			5 is in the sequence for the sum of all primes <= 5 is 10, and 10 has no prime divisor > 5.
17 is not in the sequence for the corresponding sum is 58 which has a prime divisor > 17.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A007504, A013916, A013917, A013918, A046731.

p = s = 2; lst = {}; While[p < 1000, If[ FactorInteger[s][[-1, 1]] <= p, AppendTo[lst, p]]; p = NextPrime@ p; s = s + p]; lst (* Robert G. Wilson v, Feb 09 2017 *)

isok(n) = isprime(n) && (vecmax(factor(sum(k=1, primepi(n), prime(k)))[,1]) <= n); \\ Michel Marcus, Feb 12 2017

A309718 Numbers k such that the sums of the first k and k+2 primes are also prime.

Original entry on oeis.org

2, 4, 12, 100, 122, 130, 204, 206, 214, 326, 328, 330, 332, 356, 458, 1024, 1148, 1190, 1418, 1474, 1476, 1500, 1524, 1630, 1842, 1948, 2128, 2130, 2184, 2436, 2448, 2536, 2686, 2688, 2784, 2796, 2898, 2980, 3112, 3562, 3682, 3806, 3936, 3944, 4114, 4318, 4332, 4364, 4376, 4412

Offset: 1

Philip Mizzi, Aug 13 2019

nonn

The first run of four consecutive terms that differ by two, starts at k = 326.

The first run of five consecutive terms that differ by two, starts at k = 1195374. - Daniel Suteu, Aug 17 2019

			For k=2, the sum of the first two primes 2+3 = 5 is prime. Additionally, the sum of the first k + 2=4 primes 2+3+5+7 = 17 is prime. So 2 is a term.
For k=6, the sum of the first six primes 2+3+5+7+11+13 = 41 is prime. But, the sum of the first k+2=8 primes 2+3+5+7+11+13+17+19 = 77 is not a prime. So 6 is not a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A013916.

P:= [seq(ithprime(i),i=1..10002)]:
S:= ListTools:-PartialSums(P):
select(k -> isprime(S[k]) and isprime(S[k+2]),2*[$1..5000]); # Robert Israel, Sep 01 2019

isspp(n) = isprime(sum(i=1, n, prime(i))); \\ A013916
isok(n) = isspp(n) && isspp(n+2); \\ Michel Marcus, Aug 14 2019

More terms from Michel Marcus, Aug 14 2019

A329539 Numbers m such that the sum of the first m primes as well as the sum of the squares and the sum of the cubes of the first m primes are all prime.

Original entry on oeis.org

3618, 5840, 7716, 17502, 19460, 22398, 23520, 26852, 33824, 41202, 45848, 47328, 62138, 72950, 82722, 101084, 118062, 127160, 128784, 134012, 136380, 148940, 165240, 173658, 175220, 175310, 177516, 187556, 193692, 203310, 230802, 234032, 279102, 281754, 285518, 289970, 295196, 298652

Offset: 1

Michel Marcus, Nov 16 2019

nonn

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 3618
Carlos Rivera, Puzzle 978. Improve this curio, Prime Puzzles and Problems Connection.

Cf. A013916, A098561, A124225.

Module[{nn=300000,prs,m1,m2,m3},prs=Prime[Range[nn]];m1=Accumulate[ prs];m2 = Accumulate[prs^2];m3=Accumulate[prs^3];Position[Thread[ {m1,m2,m3}],? (Total[ Boole[ PrimeQ[#]]]==3&)]]//Flatten (* _Harvey P. Dale, Jul 28 2021 *)

s=0; t=0; u=0; n=0; forprime(p=2, 1e6, s+=p; t+=p^2; u+=p^3; n++; if(isprime(u) && isprime(t) && isprime(s), print1(n, ", ")))

Name (description) modified by Harvey P. Dale, Jul 28 2021

A287748 If A070281(n) is not 0, then a(n) is index of the starting primes for form A070281(n), or a(n) = 0 if A070281(n) = 0.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 7, 0, 2, 0, 3, 2, 10, 2, 2, 0, 2, 0, 5, 0, 4, 0, 4, 0, 3, 0, 4, 0, 6, 0, 6, 0, 4, 0, 3, 0, 3, 0, 6, 0, 4, 0, 4, 0, 4, 0, 4, 0, 5, 0, 7, 0, 2, 0, 2, 0, 25, 0, 10, 2, 2, 0, 6, 2, 2, 0, 8, 0, 8, 0, 2, 0, 3, 0, 2, 0, 9, 0, 4, 0, 5, 0, 16, 0, 11

Offset: 1

XU Pingya, May 31 2017

nonn

			A070281(13) = 691 = prime(10) + ... + prime(22), thus a(13) = 10.

Cf. A070281, A013916.

cp's OEIS Frontend

A235431 The smallest positive number that must be added to or subtracted from the sum of the first n primes in order to get a prime.

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

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A277123 Numbers k such that 1 + Sum_{j=1..k} prime(j)^2 is prime.

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A280536 Numbers n such that the sum of the first n primes reduced by some power of 2 is prime.

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A282246 Primes p such that the sum of all primes <= p has no prime divisor > p.

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A309718 Numbers k such that the sums of the first k and k+2 primes are also prime.

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Maple

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A329539 Numbers m such that the sum of the first m primes as well as the sum of the squares and the sum of the cubes of the first m primes are all prime.

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Mathematica

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A287748 If A070281(n) is not 0, then a(n) is index of the starting primes for form A070281(n), or a(n) = 0 if A070281(n) = 0.

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