A072225 -id:A072225 - cp's OEIS frontend

A386275 First term of the first occurrence of a run of exactly n consecutive terms in A072225.

13, 22, 3, 137, 7, 5454, 31076, 8744076, 697642916, 23169509240, 29165083170, 10658896243375

Offset: 1

Kevin Ryde, Jul 25 2025

Exactly n means the run is maximal in the sense that it has no further consecutive term before or after.
Prime Puzzles 798 (see links) considers where runs of consecutive terms occur in A072225 (and similar) and terms here through to n=11 are per row n=3, and extra i with g(3,i)=11, of the minimal i table by Vladimir Chirkov.
a(13) and beyond are > 10658896243375 = a(12).

			For n=4, the first run of 4 consecutive terms in A072225 begins at its term 137 so that a(4) = 137,
  A072225 = ..., 134, 137, 138, 139, 140, 144
                      \----------------/
                      n=4 consecutive maximal run
The n+2 = 6 primes at prime(137) onwards are 773, 787, 797, 809, 811, 821 and the sum of any consecutive 3 of them is a prime.

Carlos Rivera, Puzzle 798. A nice puzzle by Kamenetski, The Prime Puzzles & Problems Connection.
Kevin Ryde, C Code

Cf. A072225.

C
```
/* See links. */
```

A062391 a(1) = 3, a(2) = 5; a(n+1) = smallest prime number > a(n) such that the sum of any three consecutive terms is a prime.

Original entry on oeis.org

3, 5, 11, 13, 17, 23, 31, 43, 53, 61, 67, 71, 73, 79, 89, 101, 103, 107, 127, 139, 167, 173, 181, 193, 197, 211, 223, 227, 233, 241, 269, 277, 281, 349, 353, 359, 379, 433, 467, 499, 521, 523, 557, 577, 587, 613, 631, 743, 757, 769, 821, 827, 829, 883, 947

Offset: 1

Amarnath Murthy, Jun 27 2001

What is the longest string of consecutive primes? A derived sequence could be the start of the first occurrence of a string of n consecutive primes in this sequence.

See A072225 for relevant info and links. - Zak Seidov, Sep 14 2016

			After 43, the next term is 53, since 31 + 43 + 47 = 121 is not prime and 31 + 43 + 53 = 127 is prime.

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

Cf. A072536, A072537, A000040.

Cf. A072225.

a=3; b=5; lst={a, b}; Do[c=a+b+n; If[PrimeQ[c]&&n>b&&PrimeQ[n], AppendTo[lst, n]; a=b; b=n], {n, 0, 8!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
nxt[{a_,b_}]:=Module[{p=NextPrime[b]},While[!PrimeQ[a+b+p],p= NextPrime[ p]];{b,p}]; Transpose[NestList[nxt,{3,5},70]][[1]] (* Harvey P. Dale, Aug 05 2013 *)

{ n=a1=0; forprime (p=3, 5*10^5, if (p<6 || isprime(p + s), write("b062391.txt", n++, " ", p); s=a1 + p; a1=p; if (n==1000, break)) ) } \\ Harry J. Smith, Aug 07 2009

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 02 2001

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 *)

A288041 Numbers k such that prime(k) + prime(k+1) + ... + prime(k+4) is prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 11, 14, 16, 17, 19, 21, 22, 28, 29, 30, 31, 33, 35, 36, 37, 38, 41, 43, 47, 48, 50, 56, 57, 63, 64, 70, 71, 72, 75, 76, 79, 80, 81, 84, 86, 87, 89, 91, 92, 98, 99, 100, 102, 104, 105, 106, 109, 112, 114, 119, 123, 125, 134, 140, 141, 142, 146, 148, 149, 150

Offset: 1

Zak Seidov, Jun 04 2017

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000720 (PrimePi), A072225 (numbers n such that prime(n) + prime(n+1) + prime(n+2) is prime), A073681 (smallest of three consecutive primes whose sum is a prime), A152468 (smallest of five consecutive primes whose sum is a prime).

With[{nn = 154}, Function[s, Select[Range[nn - 4], PrimeQ@ Total@ Take[s, {#, # + 4}] &]]@ Prime@ Range@ nn] (* Michael De Vlieger, Jun 06 2017 *)
Position[Total/@Partition[Prime[Range[200]],5,1],?(PrimeQ[#]&)]//Flatten (* _Harvey P. Dale, Sep 09 2024 *)

list(lim)=my(v=List(),u=primes(5),n=1); forprime(p=13,, if(n++>lim, break); u=concat(u[2..5],p); if(isprime(vecsum(u)), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 10 2017

a(n) = pi(A152468(n)) = A000720(A152468(n)).

A117675 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime and also there is a j such that prime(j) + prime(j+1) + prime(j+2) = prime(k).

Original entry on oeis.org

9, 11, 13, 20, 23, 29, 47, 64, 70, 88, 121, 126, 145, 148, 153, 174, 190, 195, 201, 213, 223, 245, 294, 298, 320, 337, 369, 381, 429, 436, 445, 462, 486, 495, 504, 536, 548, 584, 596, 608, 639, 677, 747, 819, 827, 831, 868, 877, 887, 902, 905, 970

Offset: 1

Roger L. Bagula, Apr 12 2006

nonn

Cf. A109756, A072225.

isok2(k)={my(q=prime(k), p=q\3); while(p>2, p=precprime(p-1); my(p2=nextprime(p+1), t=p+p2+nextprime(p2+1)); if(t<=q, return(t==q))); 0}
isok(k)={my(p1=prime(k), p2=nextprime(p1+1), p3=nextprime(p2+1)); isprime(p1+p2+p3) && isok2(k)}
select(isok, [1..1000]) \\ Andrew Howroyd, Jul 23 2024

Edited and more terms from Andrew Howroyd, Jul 23 2024

A117714 a(n) = (A034962(n) - A152470(n))/2.

Original entry on oeis.org

6, 9, 12, 18, 21, 26, 30, 34, 42, 56, 64, 69, 72, 81, 86, 102, 111, 144, 150, 160, 165, 198, 217, 231, 274, 282, 288, 300, 312, 342, 348, 351, 381, 393, 405, 414, 432, 453, 459, 465, 473, 495, 501, 515

Offset: 1

Roger L. Bagula, Apr 13 2006

The sequence is always increasing.

Cf. A072225, A073681.

a = Flatten[Table[If[PrimeQ[Prime[n] + Prime[n + 1] + Prime[n + 2]] == True, If [Prime[n] + Prime[n + 1] + Prime[n + 2] - Prime[m] == 0, {( Prime[m] - Prime[n + 2])/2}, {}], {}], {n, 1, 100}, {m, 1, 500}]]

Description simplified by the Assoc. Eds. of the OEIS, Jun 27 2010

A185313 Start of a sequence of n consecutive primes such that the sum of any three consecutive members is also prime.

Original entry on oeis.org

2, 2, 5, 5, 5, 17, 17, 53507, 364187, 155650237, 15644021363, 604394270371, 767783880089

Offset: 1

Charles R Greathouse IV, Feb 08 2012

a(14) > 1.8*10^14. - Giovanni Resta, Jun 07 2017

			Vacuously, the sum of every three consecutive members of {2, 3} is prime, so a(2) = 2.  a(4) = 5 because 5 + 7 + 11 and 7 + 11 + 13 are prime.

Cf. A072225.

a(n)=if(n<3,return(2),n-=2);my(len=0,p=2,q=3);forprime(r=5,default(primelimit),if(isprime(p+q+r),if(len++==n,my(t=p);for(i=2,n,t=precprime(t-1));return(t)),len=0);p=q;q=r) \\ Charles R Greathouse IV, Feb 08 2012

a(11) from Charles R Greathouse IV, Feb 10 2012
a(12) from Charles R Greathouse IV, Feb 12 2012
a(13) from Zak Seidov, Jun 03 2017

A260907 Numbers n such that prime(n) + prime(n+1) + prime(n+2) is not a prime.

Original entry on oeis.org

1, 2, 6, 12, 14, 15, 17, 21, 24, 25, 27, 28, 30, 31, 32, 33, 36, 39, 40, 41, 42, 43, 44, 46, 48, 49, 51, 52, 53, 54, 55, 56, 57, 59, 63, 65, 66, 67, 71, 72, 73, 74, 76, 78, 81, 82, 84, 85, 86, 89, 92, 93, 96, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108

Offset: 1

Vincenzo Librandi, Nov 18 2015

Complement of A072225.

			6 is in the sequence because prime(6) + prime(7) + prime(8) = 13 + 17 + 19 = 49 is not a prime.

Cf. A000040, A072225, A174742 (associated primes).

[n: n in [1..200] | not IsPrime(NthPrime(n) + NthPrime(n+1) + NthPrime(n+2))];

Select[Range[200], !PrimeQ[Prime[#] + Prime[# + 1] + Prime[# + 2]] &]

for(n=1, 1e2, if(!isprime(prime(n)+prime(n+1)+prime(n+2)), print1(n, ", "))) \\ Altug Alkan, Nov 19 2015

[n for n in (1..200) if not is_prime(nth_prime(n) + nth_prime(n+1) + nth_prime(n+2))] # Bruno Berselli, Nov 19 2015

cp's OEIS Frontend

A386275 First term of the first occurrence of a run of exactly n consecutive terms in A072225.

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A062391 a(1) = 3, a(2) = 5; a(n+1) = smallest prime number > a(n) such that the sum of any three consecutive terms is a prime.

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

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Maple

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

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Maple

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A288041 Numbers k such that prime(k) + prime(k+1) + ... + prime(k+4) is prime.

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A117675 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime and also there is a j such that prime(j) + prime(j+1) + prime(j+2) = prime(k).

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A117714 a(n) = (A034962(n) - A152470(n))/2.

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A185313 Start of a sequence of n consecutive primes such that the sum of any three consecutive members is also prime.

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