A073681 -id:A073681 - cp's OEIS frontend

A034962 Primes that are the sum of three consecutive primes.

23, 31, 41, 59, 71, 83, 97, 109, 131, 173, 199, 211, 223, 251, 269, 311, 349, 439, 457, 487, 503, 607, 661, 701, 829, 857, 883, 911, 941, 1033, 1049, 1061, 1151, 1187, 1229, 1249, 1303, 1367, 1381, 1409, 1433, 1493, 1511, 1553, 1667, 1867, 1931, 1973, 1993

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

Or, primes in A034961 (Sums of three consecutive primes). - Zak Seidov, Feb 16 2011

			E.g., 131 = 41 + 43 + 47.
A034962(n) = p+q+r, where p = A073681(n), and p<q<r are three consecutive primes. - _Zak Seidov_, Mar 09 2009

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Table of n, primepi(A073681(n)), primepi(A034962(n)), A073681(n), A152469(n), A152470(n), A034962(n) for n = 1..10000
Zak Seidov, Table of a(1000*k) for k=1..1346.

Cf. A001043, A011974, A034707, A034961. Different from A050207.

Cf. A073681 (smallest of three consecutive primes whose sum is a prime).

[a: n in [1..150] | IsPrime(a) where a is NthPrime(n)+NthPrime(n+1)+NthPrime(n+2)]; // Vincenzo Librandi, Jun 23 2016

a:=proc(n) if isprime(ithprime(n)+ithprime(n+1)+ithprime(n+2))=true then ithprime(n)+ithprime(n+1)+ithprime(n+2) else fi end: seq(a(n), n=1..120); # Emeric Deutsch, Apr 24 2006

a = {}; Do[k = Prime[x] + Prime[x + 1] + Prime[x + 2]; If[PrimeQ[k], AppendTo[a, k]], {x, 1, 350}]; a (* Artur Jasinski, Jan 27 2007 *)
Select[(Plus@@@Partition[Prime[Range[200]],3,1]),PrimeQ] (* Zak Seidov, Feb 07 2012 *)
Select[ListConvolve[{1,1,1},Prime[Range[200]]],PrimeQ] (* Harvey P. Dale, Jul 12 2013 *)

forprime(p=2,1000, p2=nextprime(p+1); p3=nextprime(p2+1); q=p+p2+p3; if(isprime(q),print1(q",")) ) \\ Max Alekseyev, Jan 26 2007

{p=2;q=3;for(n=1,100,r=nextprime(q+1); if(isprime(t=p+q+r),print1(t","));p=q;q=r;)} \\ Zak Seidov, Mar 09 2009

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    p, q, r = 2, 3, 5
    while True:
        if isprime(p+q+r): yield p+q+r
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 50))) # Michael S. Branicky, Dec 27 2022

A152468 Smallest of five consecutive primes whose sum is a prime.

Original entry on oeis.org

5, 7, 11, 13, 19, 29, 31, 43, 53, 59, 67, 73, 79, 107, 109, 113, 127, 137, 149, 151, 157, 163, 179, 191, 211, 223, 229, 263, 269, 307, 311, 349, 353, 359, 379, 383, 401, 409, 419, 433, 443, 449, 461, 467, 479, 521, 523, 541, 557, 569, 571, 577, 599, 613, 619

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 05 2008

nonn

Surprisingly many terms are also in A073681. - Zak Seidov, Dec 17 2012

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A073681, A034965, A180948, A189571, A180950, A226380.

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];p3=Prime[n+3];p4=Prime[n+4];If[PrimeQ[p=p0+p1+p2+p3+p4],AppendTo[lst,p0]],{n,6!}];lst
Transpose[Select[Partition[Prime[Range[500]], 5, 1], PrimeQ[Total[#]] &]][[1]] (* Harvey P. Dale, Jun 05 2013 *)
Prime[Select[Range[150], PrimeQ[Sum[Prime[# + i], {i, 0, 4}]] &]] (* Bruno Berselli, Aug 21 2013 *)

{a=2; b=3; c=5; d=7; e=11; for(n=1,100, s=a+b+c+d+e;
if(isprime(s), print1(a", ")); a=b; b=c; c=d; d=e; e=nextprime(e+2))} /* Zak Seidov, Dec 17 2012 */

More cross references from Harvey P. Dale, Jun 05 2013

A072225 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 18, 19, 20, 22, 23, 26, 29, 34, 35, 37, 38, 45, 47, 50, 58, 60, 61, 62, 64, 68, 69, 70, 75, 77, 79, 80, 83, 87, 88, 90, 91, 94, 95, 97, 101, 113, 116, 119, 120, 121, 126, 128, 132, 133, 134

Offset: 1

Joseph L. Pe, Jul 04 2002

The sequence contains 298884 terms <= 2000000, so nearly 15% of the numbers <= 2000000 are in the sequence. - Dmitry Kamenetsky, Aug 08 2015
Heuristically, we might expect the "probability" of n being in the sequence to be on the order of 1/log(n). - Robert Israel, Aug 09 2015
The first 8 consecutive integers in this sequence start from 8744076. - Dmitry Kamenetsky, Aug 28 2015
The first 9 consecutive integers in this sequence start from 697642916. - Dmitry Kamenetsky, Sep 08 2015
Vladimir Chirkov found that the first 10 and 11 consecutive integers in this sequence start from 23169509240 and 29165083170, respectively (see The Prime Puzzles link and A386275). - Dmitry Kamenetsky, Sep 08 2015

			9 is in the sequence because prime(9) + prime(10) + prime(11) = 23 + 29 + 31 = 83 is a prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 798. A nice puzzle by Kamenetski, The Prime Puzzles & Problems Connection.

Cf. A034962, A073681, A117673, A152469, A152470.

Cf. A386275 (first occurrences of runs).

[n: n in [0..600]| IsPrime(NthPrime(n)+NthPrime(n+1)+NthPrime(n+2))]; // Vincenzo Librandi, Apr 06 2011

a:=proc(n) if isprime(ithprime(n)+ithprime(n+1)+ithprime(n+2))=true then n else fi end: seq(a(n),n=1..150); # Emeric Deutsch, Apr 24 2006

Select[Range[10^4], PrimeQ[Prime[ # ] + Prime[ # + 1] + Prime[ # + 2]] &]

isok(n)=isprime(prime(n)+prime(n+1)+prime(n+2)) \\ Anders Hellström, Aug 20 2015

A180948 Smallest of seven (7) consecutive primes whose sum is a prime.

Original entry on oeis.org

17, 19, 23, 29, 31, 43, 47, 53, 61, 71, 79, 89, 97, 103, 107, 113, 127, 137, 151, 233, 257, 313, 317, 359, 367, 373, 379, 383, 401, 461, 463, 487, 499, 503, 509, 521, 577, 587, 617, 619, 761, 797, 821, 827, 839, 853, 881, 883, 907, 1019, 1061, 1063, 1069, 1097

Offset: 1

Carmine Suriano, Sep 27 2010

nonn

There are twins such as (17,19); (461,463); (1061,1063).

There are also consecutives such as (17,19,23,29,31); (359,367,373,379,383); (1949,1951,1973).

			a(7)=47+53+59+61+67+71+73=431 is a prime.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A073681, A082246, A152468, A189571, A180950, A226380.

Transpose[Select[Partition[Prime[Range[500]],7,1],PrimeQ[Total[#]]&]] [[1]] (* Harvey P. Dale, Jun 05 2013 *)

More cross references from Harvey P. Dale, Jun 05 2013

A180950 Smallest prime such that the sum of successive 11 primes is a prime.

Original entry on oeis.org

5, 7, 11, 13, 19, 23, 37, 41, 59, 101, 103, 107, 109, 113, 137, 149, 151, 157, 167, 173, 191, 269, 281, 283, 293, 307, 331, 349, 353, 359, 421, 431, 463, 479, 487, 499, 503, 569, 599, 607, 613, 617, 619, 677, 733, 761, 773, 823, 853, 883, 967, 977, 1051, 1087

Offset: 1

Carmine Suriano, Sep 27 2010

nonn

Sum i=0 to 10 of prime(n+i) is a prime.
There are many twins: (11,13); (149,151); (1301,1303); (1997,1999); (8969,8971) ...
There are also consecutive primes: (101,103,107,109,113); (607,613,617,619); (8681,8689,8693)

			a(5)=19 since 19+23+29+31+37+41+43+47+53+59+61=443 is a prime.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A127338, A073681, A152468, A180948, A189571, A226380.

Transpose[Select[Partition[Prime[Range[500]], 11, 1], PrimeQ[Total[#]] &]][[1]] [[1]] (* Harvey P. Dale, Jun 05 2013 *)
Prime[Select[Range[200], PrimeQ[Sum[Prime[# + i], {i, 0, 10}]] &]] (* Bruno Berselli, Aug 22 2013 *)

More cross references from Harvey P. Dale, Jun 05 2013

A226380 Smallest of 101 consecutive primes whose sum is prime.

Original entry on oeis.org

83, 89, 139, 179, 181, 277, 281, 353, 409, 479, 499, 521, 571, 587, 643, 727, 839, 883, 887, 919, 929, 971, 977, 1019, 1021, 1117, 1213, 1223, 1237, 1259, 1303, 1327, 1367, 1381, 1399, 1423, 1433, 1481, 1483, 1667, 1723, 1789, 1823, 1861, 1879, 1913, 2083

Offset: 1

Harvey P. Dale, Jun 05 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A073681, A152468, A180948, A189571, A180950.

Transpose[Select[Partition[Prime[Range[500]],101,1],PrimeQ[Total[#]]&]] [[1]]
Prime[Select[Range[400], PrimeQ[Sum[Prime[# + i], {i, 0, 100}]] &]] (* Bruno Berselli, Aug 21 2013 *)

A189571 Smallest of nine consecutive primes whose sum is a prime.

Original entry on oeis.org

3, 29, 31, 37, 47, 79, 83, 89, 107, 109, 127, 131, 139, 149, 157, 173, 179, 193, 197, 199, 211, 241, 277, 347, 359, 367, 373, 389, 397, 433, 449, 487, 491, 521, 577, 593, 619, 643, 659, 677, 743, 761, 829, 853, 953, 977, 1049, 1063, 1087, 1129, 1151, 1193

Offset: 1

Bruno Berselli, Apr 23 2011

nonn

First 7-tuple of consecutive primes belonging to the sequence: 118061, 118081, 118093, 118127, 118147, 118163, 118169. Twin primes in the sequence: 29, 31; 107, 109; 197, 199; 1427, 1429; 1607, 1609; 1721, 1723; 4019, 4021, etc. [Bruno Berselli, Aug 26 2013]

			47 is in the sequence because 47+53+59+61+67+71+73+79+83 = 593 and 593 is prime.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A073681, A152468, A180948, A180950, A226380.

[ NthPrime(n): n in [1..190] | IsPrime(&+[NthPrime(n+s): s in [0..8]]) ];

Transpose[Select[Partition[Prime[Range[500]],9,1],PrimeQ[Total[#]]&]] [[1]] (* Harvey P. Dale, Jun 05 2013 *)

from sympy import isprime, nextprime
def aupto(limit):
  plst, alst = [3, 5, 7, 11, 13, 17, 19, 23, 29], []
  while plst[0] <= limit:
    if isprime(sum(plst)): alst.append(plst[0])
    plst = plst[1:] + [nextprime(plst[-1])]
  return alst
print(aupto(1200)) # Michael S. Branicky, Mar 29 2021

Additional cross reference from Harvey P. Dale, Jun 05 2013

A152470 Largest of three consecutive primes whose sum is a prime.

Original entry on oeis.org

11, 13, 17, 23, 29, 31, 37, 41, 47, 61, 71, 73, 79, 89, 97, 107, 127, 151, 157, 167, 173, 211, 227, 239, 281, 293, 307, 311, 317, 349, 353, 359, 389, 401, 419, 421, 439, 461, 463, 479, 487, 503, 509, 523, 563, 631, 647, 661, 673, 677, 719, 733, 757, 761, 769

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 05 2008

nonn

			3+5+7 = 15 is composite.
5+7+11 = 23 is prime and (5, 7, 11) are consecutive primes so a(1) = 11.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A073681, A072225, A152468, A152469.

Primes:= select(isprime,[2,(2*i+1 $ i=1..10000)]):
Primes[select(t -> isprime(Primes[t-2]+Primes[t-1]+Primes[t]),[$3..nops(Primes)])];
# Robert Israel, Aug 29 2014

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];If[PrimeQ[p0+p1+p2],AppendTo[lst,p2]],{n,6!}];lst

s=[]; for(n=1, 1000, if(isprime(prime(n)+prime(n+1)+prime(n+2)), s=concat(s, prime(n+2)))); s \\ Colin Barker, Aug 25 2014

A152469 Second smallest of three consecutive primes whose sum is a prime.

Original entry on oeis.org

7, 11, 13, 19, 23, 29, 31, 37, 43, 59, 67, 71, 73, 83, 89, 103, 113, 149, 151, 163, 167, 199, 223, 233, 277, 283, 293, 307, 313, 347, 349, 353, 383, 397, 409, 419, 433, 457, 461, 467, 479, 499, 503, 521, 557, 619, 643, 659, 661, 673, 709, 727, 751, 757, 761

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 05 2008

nonn

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

Cf. A073681, A072225, A152468, A152470.

t0:=[];
t1:=[];
t2:=[];
t3:=[];
for i from 1 to 1000 do
t3:=ithprime(i)+ithprime(i+1)+ithprime(i+2);
if isprime(t3) then
t0:=[op(t0),i];
t1:=[op(t1),ithprime(i)];
t2:=[op(t2),ithprime(i+1)];
t3:=[op(t2),ithprime(i+2)];
fi;
od:
t2;

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];If[PrimeQ[p0+p1+p2],AppendTo[lst,p1]],{n,6!}];lst
Select[Partition[Prime[Range[200]],3,1],PrimeQ[Total[#]]&][[All,2]] (* Harvey P. Dale, May 08 2021 *)

A182121 Primes p such that the sum of both three and five consecutive primes starting with p is prime.

Original entry on oeis.org

5, 7, 11, 19, 29, 31, 53, 67, 79, 109, 149, 157, 163, 211, 229, 311, 349, 379, 401, 409, 449, 467, 653, 757, 809, 839, 857, 863, 883, 983, 997, 1033, 1087, 1103, 1187, 1193, 1289, 1301, 1303, 1409, 1481, 1523, 1553, 1637, 1663, 1669, 1709, 1951, 1973

Offset: 1

Zak Seidov, Dec 17 2012

nonn

			5 is in the sequence because 5 + 7 + 11 = 23 is prime and 5 + 7 + 11 + 13 + 17 = 53 is also prime.

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

Intersection of A073681 and A152468.

cpQ[n_]:=Module[{ppi=PrimePi[n],cnsc},cnsc=Prime[Range[ppi,ppi+4]];And@@ PrimeQ[ {Total[cnsc],Total[Take[cnsc,3]]}]]; Select[Prime[Range[300]],cpQ] (* Harvey P. Dale, Mar 28 2013 *)
Select[Partition[Prime[Range[500]],5,1],AllTrue[{Total[Take[#,3]],Total[#]},PrimeQ]&][[;;,1]] (* Harvey P. Dale, Feb 11 2024 *)

{a=2;b=3;c=5;d=7;e=11;for(n=1,300,s=a+b+c+d+e;
if(isprime(s)&&isprime(a+b+c),print1(a","));a=b;b=c;c=d;d=e;e=nextprime(e+2))}

cp's OEIS Frontend

A034962 Primes that are the sum of three consecutive primes.

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A152468 Smallest of five consecutive primes whose sum is a prime.

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A072225 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime.

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A180948 Smallest of seven (7) consecutive primes whose sum is a prime.

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A180950 Smallest prime such that the sum of successive 11 primes is a prime.

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A226380 Smallest of 101 consecutive primes whose sum is prime.

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A189571 Smallest of nine consecutive primes whose sum is a prime.

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