A013916 -id:A013916 - cp's OEIS frontend

A098563 Numbers n such that the sum of the cubes of the first n primes is prime.

4, 8, 38, 48, 98, 102, 118, 128, 130, 132, 156, 168, 172, 178, 180, 190, 202, 208, 308, 346, 358, 364, 424, 482, 540, 600, 602, 614, 646, 676, 722, 748, 768, 776, 782, 792, 838, 902, 1016, 1028, 1036, 1058, 1062, 1082, 1086, 1100, 1102, 1132, 1144, 1176

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

n must clearly be even.

			4 is a term as the sum of the cubes of the first four primes is 2^3 + 3^3 + 5^3 + 7^3 = 503, which is prime.

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Cf. A066525 (corresponding primes), A098561 (sums of squares of primes), A013916 (sums of primes), A098999 (sums of cubes of primes).

with(numtheory): P:=proc(n) add(ithprime(k)^3, k=1..n): end:
A098563 := proc(n)local m: option remember: if(n=0)then return 0: fi: m:=procname(n-1)+2: while true do if(isprime(P(m)))then return m:fi: m:=m+2:od: end:
seq(A098563(n), n=1..50); # Nathaniel Johnston, Apr 21 2011

Select[Range[1000], PrimeQ[Sum[Prime[i]^3, {i, #}]] &] (* Carl Najafi, Aug 22 2011 *)

lista(nn) = {s = 0; ip = 0; forprime (p=1, nn, ip++; if (isprime(s+=p^3), print1(ip, ", ")););} \\ Michel Marcus, Aug 22 2015

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 *)

A046280 Numbers k such that the sum of the first k lucky numbers, A046279(k), is prime.

Original entry on oeis.org

3, 15, 25, 31, 37, 55, 57, 85, 91, 99, 133, 153, 173, 179, 183, 219, 293, 301, 305, 353, 355, 383, 387, 389, 393, 397, 403, 407, 409, 427, 453, 467, 491, 495, 515, 517, 519, 547, 573, 681, 691, 703, 705, 723, 731, 733, 747, 761, 795, 829, 837, 845, 865, 873

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

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

Cf. A013916, A000959, A046279, A046281.

Offset corrected by Amiram Eldar, Nov 16 2019

A102863 a(n)=1 if at least one of the first n primes is a divisor of the sum of the first n primes; otherwise a(n)=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1

Offset: 1

Giovanni Teofilatto, Mar 01 2005

a(n) = 0 if and only if n is in A013916. - Robert Israel, Jan 04 2017

			a(2)=0 because none of the first 2 primes (2, 3) is a divisor of 2+3; a(5)=1 because among the first 5 primes (namely, 2,3,5,7,11) there are divisors of 2+3+5+7+11=28.

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

A105783(n) gives number of primes among the first n primes that are divisors of the sum of the first n primes.

Cf. A013916, A136443.

with(numtheory):
a:=proc(n)
   if nops(factorset(sum(ithprime(k),k=1..n)) intersect {seq(ithprime(j),j=1..n)}) >0 then
      1
   else
      0
   fi
end:
seq(a(n),n=1..130); # Emeric Deutsch
# alternative:
N:= 500: # to get the first N terms
A:= Vector(N):
S:= 2: P:= 2: p:= 2: A[1]:= 1:
for n from 2 to N do
  p:= nextprime(p);
  S:= S+p; P:= P*p;
  if igcd(S,P) > 1 then A[n]:= 1 fi
od:
convert(A,list); # Robert Israel, Jan 04 2017

a[n_] := Module[{pp = Prime[Range[n]], t}, t = Total[pp]; Boole[AnyTrue[pp, Divisible[t, #]&]]];
Array[a, 100] (* Jean-François Alcover, Jun 16 2020 *)

Edited and extended by Emeric Deutsch, Apr 19 2005

A124225 Numbers n such that the sum of the first n primes is prime and the sum of the squares of the first n primes is also prime.

Original entry on oeis.org

2, 158, 192, 216, 356, 426, 548, 680, 1178, 1196, 1466, 1500, 1524, 2324, 2438, 2904, 2990, 3060, 3146, 3618, 3902, 4110, 4134, 4346, 4602, 5790, 5840, 6186, 6344, 6710, 6720, 6836, 6990, 7592, 7632, 7716, 7790, 7838, 8156, 8420, 8622, 8658, 8664, 9092

Offset: 1

Tanya Khovanova, Dec 13 2006

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 3618
Carlos Rivera, Puzzle 978. Improve this curio, Prime Puzzles and Problems Connection.

Intersection of A098561 (Numbers n such that the sum of the squares of the first n primes is prime) and A013916 (Numbers n such that the sum of the first n primes is prime).

a013916:=func; a098561:=func; [n: n in [1..10000]|a013916(n) and a098561(n)];  // Bruno Berselli, Dec 28 2011

With[{nn=9100},Position[Thread[{Accumulate[Prime[Range[nn]]],Accumulate[ Prime[ Range[ nn]]^2]}],?(PrimeQ[ #[[1]]]&&PrimeQ[#[[2]]]&),1,Heads-> False]]//Flatten (* _Harvey P. Dale, Aug 18 2020 *)

s=0;t=0;n=0;forprime(p=2,1e6,s+=p;t+=p^2;n++;if(isprime(t)&&isprime(s),print1(n", "))) \\ Charles R Greathouse IV, Dec 28 2011

More terms from Bruno Berselli, Dec 28 2011

Definition rephrased by Harvey P. Dale, Aug 18 2020

A213650 Numbers k such that the sum of the first k primes is semiprime.

Original entry on oeis.org

3, 7, 8, 10, 16, 18, 22, 28, 32, 34, 36, 38, 44, 46, 48, 54, 55, 58, 59, 65, 66, 72, 75, 82, 92, 93, 94, 104, 106, 110, 118, 120, 133, 136, 137, 138, 140, 141, 142, 144, 148, 150, 154, 156, 164, 168, 170, 174, 190, 194, 202, 210, 212, 218, 224, 226, 232, 234

Offset: 1

Michel Lagneau, Jun 17 2012

nonn

Numbers k such that A007504(k) is included in A001358.

			8 is in the sequence because the sum of the first 8 primes is  2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77 = 7*11, which is semiprime.

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

Cf. A001358, A007504, A013916, A092189 (numbers n such that sum of first n semiprimes is a semiprime), A092190 (semiprimes that are the sum of first n semiprimes for some n), A180152 (numbers n such that the sum of the first n semiprimes is a prime).

with(numtheory): for n from 1 to 500 do:s:=sum(‘ithprime(k)’, ’k’=1..n):if bigomega(s)=2 then printf(`%d, `, n):else fi:od:

Flatten[Position[Accumulate[Prime[Range[300]]],_?(PrimeOmega[#]==2&)]]

isok(n) = bigomega(vecsum(primes(n))) == 2; \\ Michel Marcus, Sep 18 2017

A321342 Numbers k such that if j is the sum of the first k primes, then the sum of the first j primes is prime.

Original entry on oeis.org

1, 9, 15, 19, 73, 85, 87, 103, 121, 157, 175, 277, 313, 317, 341, 357, 375, 385, 391, 421, 443, 447, 523, 525, 539, 571, 607, 611, 645, 701, 779, 783, 791, 799, 823, 831, 835, 853, 889, 907, 911, 925, 977, 1051, 1075, 1081, 1087, 1095, 1117, 1125, 1135, 1157, 1181, 1187, 1223, 1257, 1305, 1325

Offset: 1

David James Sycamore, Nov 06 2018

nonn

k is in the sequence if A007504(j) is prime, where j = A007504(k). A007504(j) must be odd to be prime, so j must be even and k must be odd. Therefore all terms are odd. The subsequence of primes is A321343.

			A007504(1) = 2 and A007504(2) = 5, a prime therefore 1 is a term.
A007504(3) = 10 and A007504(10) = 129, not prime, therefore 3 is not a term.
A007504(9) = 100 and A007504(100) = 24133, a prime so 9 is a term.

Ray Chandler, Table of n, a(n) for n = 1..16000
Daniel Suteu, Perl program

Cf. A007504, A013916, A321343.

N:=2000:
for n from 1 to N by 2 do
X:=add(ithprime(r),r=1..n);
Y:=add(ithprime(k),k=1..X);
if isprime(Y) then print(n);
end if:
end do:

primeSum[n_] := Sum[Prime[i], {i, n}]; Select[Range[300], PrimeQ[primeSum[primeSum[#]]] &] (* Amiram Eldar, Nov 07 2018 *)

sfp(n) = sum(k=1, n, prime(k)); \\ A007504
isok(n) = isprime(sfp(sfp(n))); \\ Michel Marcus, Nov 08 2018

A321343 Primes p such that if k is the sum of the first p primes then the sum of the first k primes is prime.

Original entry on oeis.org

19, 73, 103, 157, 277, 313, 317, 421, 443, 523, 571, 607, 701, 823, 853, 907, 911, 977, 1051, 1087, 1117, 1181, 1187, 1223, 1451, 1453, 1531, 1667, 1861, 2551, 2999, 3169, 3257, 3389, 3583, 3671, 3889, 3907, 3911, 4597, 4691, 4919, 5347, 5527, 5569, 5623, 5657, 5839

Offset: 1

David James Sycamore, Nov 06 2018

nonn

Primes p such that A007504(A007504(p)) is prime; subsequence of A321342.

			The smallest prime p such that A007504(p) is prime is 19 (sum of first 19 primes is 100 and sum of first 100 primes is 24133, which is prime). Therefore a(1) = 19.

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

Cf. A007504, A013916, A321439, A321342.

N:=2000:
for n from 1 to N by 2 do
X:=add(ithprime(k),k=1..n);
Y:=add(ithprime(j),j=1..X);
if isprime(n)and isprime(Y) then print(n);
end if:
end do:

primeSum[n_] := Sum[Prime[i], {i, n}]; Select[Range[300], PrimeQ[#] && PrimeQ[primeSum[primeSum[#]]] &] (* Amiram Eldar, Nov 07 2018 *)

upto(n) = {my(v = vector(n+1), res = List, t = 1, setv, s = 0, Ap = 0, AAp=0, q =0); v[1] = 2; forprime(p = 3, prime(n+1), t++; v[t] = v[t-1] + p); t=1; vt = v[1]; forprime(p = 2, , AAp += p; q++; if(q == vt, if(isprime(t) && isprime(AAp), listput(res, t); print1(t", ")); t++; if(t>=n, return(res)); vt = v[t])); res} \\ David A. Corneth, Nov 09 2018

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

A366092 Distance from the sum of the first n primes to the nearest prime.

Original entry on oeis.org

2, 0, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 0, 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, 0, 11, 2, 1, 0, 3, 2, 3, 2, 7, 2, 1, 2, 3, 4, 3, 2, 3, 4, 5, 2, 5, 4, 3, 10, 3

Offset: 0

Paolo Xausa, Sep 29 2023

Positions of zeros are given by A013916.

Positions of records are given by A366093.

			a(3) = 1 because the sum of the first 3 primes is 2 + 3 + 5 = 10, the nearest prime is 11 and 11 - 10 = 1.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000040, A007504, A013916, A051699, A366093, A366094.

Cf. also A351830, A354330.

pDist[n_]:=If[PrimeQ[n],0,Min[NextPrime[n]-n,n-NextPrime[n,-1]]];
A366092list[nmax_]:=Map[pDist,Prepend[Accumulate[Prime[Range[nmax]]],0]];
A366092list[100]

from sympy import prime, nextprime, prevprime
def A366092(n): return min((m:=sum(prime(i) for i in range(1,n+1)))-prevprime(m+1),nextprime(m)-m) if n else 2 # Chai Wah Wu, Oct 03 2023

a(n) = A051699(A007504(n)).

a(n) = abs(A007504(n) - A366094(n)).

A020641 a(n)-th prime is sum of first k primes for some k.

Original entry on oeis.org

1, 3, 7, 13, 45, 60, 977, 1108, 2470, 2687, 2784, 3126, 3470, 3977, 4100, 4511, 4644, 5668, 6148, 6627, 6963, 8407, 9767, 10379, 11007, 11220, 11449, 12111, 12332, 23080, 25001, 28009, 28357, 28709, 29060, 29404, 30824, 32271, 33397, 33764, 47735, 52278

Offset: 1

N. J. A. Sloane, Renaud Lifchitz (100637.64(AT)CompuServe.COM), David W. Wilson

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A013916, A013917, A013918.

PrimePi /@ Select[ FoldList[Plus, 0, Prime@ Range@450], PrimeQ@# &] (* Robert G. Wilson v Sep 28 2006 *)
PrimePi/@Select[Accumulate[Prime[Range[500]]],PrimeQ] (* Harvey P. Dale, Jun 05 2013 *)

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A098563 Numbers n such that the sum of the cubes of the first n primes is 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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A046280 Numbers k such that the sum of the first k lucky numbers, A046279(k), is prime.

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A102863 a(n)=1 if at least one of the first n primes is a divisor of the sum of the first n primes; otherwise a(n)=0.

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A124225 Numbers n such that the sum of the first n primes is prime and the sum of the squares of the first n primes is also prime.

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A213650 Numbers k such that the sum of the first k primes is semiprime.

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A321342 Numbers k such that if j is the sum of the first k primes, then the sum of the first j primes is prime.

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A321343 Primes p such that if k is the sum of the first p primes then the sum of the first k primes is prime.

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