This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
251, 257, 263, 269 are consecutive primes: 257 = 251 + 6, 263 = 251 + 12, 269 = 251 + 18.
N:=10^5: # to get all terms <= N. Primes:=select(isprime,[seq(i,i=3..N+18,2)]): Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1], Primes[t+3]-Primes[t+2]]=[6,6,6], [$1..nops(Primes)-3])]; # Muniru A Asiru, Aug 04 2017
A033451 = Reap[ For[p = 2, p < 100000, p = NextPrime[p], p2 = NextPrime[p]; If[p2 - p == 6, p3 = NextPrime[p2]; If[p3 - p2 == 6, p4 = NextPrime[p3]; If[p4 - p3 == 6, Sow[p]]]]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012 *) Transpose[Select[Partition[Prime[Range[16000]],4,1],Union[ Differences[ #]] == {6}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)
p=2;q=3;r=5;forprime(s=7,1e4,if(s-p==18 && s-q==12 && s-r==6, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Feb 14 2013
p(271)=1741: 1741,1747,1753,1759 are consecutive primes,1747=1741+6,1753=1741+12,1759=1741+18
p[n_]:=Prime[n];Select[Range[15000],p[ #+1]-p[ # ]==p[ #+2]-p[ #+1]==p[ #+3]-p[ #+2]==6&] (* Zak Seidov, Mar 05 2006 *) PrimePi[#[[1]]]&/@Select[Partition[Prime[Range[11000]],4,1],Differences[#]=={6,6,6}&] (* Harvey P. Dale, Oct 28 2023 *)
6*41+5=251, 6*41+11=257, 6*41+17=263, 6*41+23=269; 251,257,263,269 are consecutive primes.
Select[(#-5)/6&/@Transpose[Select[Partition[Prime[Range[11500]],4,1], Union[Differences[#]]=={6}&]][[1]],IntegerQ] (* Harvey P. Dale, Nov 18 2013 *)
Select[Range[20000],(PrimeQ[6#+5]&&Differences[NestList[ NextPrime[ #]&, 6 #+5,3]] =={6,6,6})||(PrimeQ[6#+1]&&Differences[NestList[NextPrime[#]&,6 #+1,3]]=={6,6,6})&] (* Harvey P. Dale, Sep 23 2016 *)
p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==18&&s-q==12&&s-r==6,print1(p\6", "));p=q;q=r;r=s) \\ Charles R Greathouse IV, Dec 27 2011
251,257,263,269 are consecutive primes,257=251+6,263=251+12,269=251+18 and 251=6*41+5
Transpose[Select[Partition[Prime[Range[30000]],4,1],Differences[#]=={6,6,6}&&IntegerQ[(#[[1]]-5)/6]&]][[1]] (* Harvey P. Dale, Dec 12 2015 *)
1741, 1747, 1753, 1759 are consecutive primes, 1747 = 1741 + 6, 1753 = 1741 + 12, 1759 = 1741 + 18 and 1741 = 6 * 290 + 1.
filter:= p -> isprime(p) and nextprime(p) = p+6 and nextprime(p+6)=p+12 and nextprime(p+12)=p+18: select(filter, [seq(i,i=1..10^6,6)]); # Robert Israel, Nov 11 2020
isok(p) = my(q,r,s); isprime(p) && ((p % 6) == 1) && ((q=nextprime(p+1)) == p+6) && ((r=nextprime(q+1)) == p+12) && ((s=nextprime(r+1)) == p+18); \\ Michel Marcus, Sep 20 2019
p(271)=1741: 1741,1747,1753,1759 are consecutive primes,1747=1741+6,1753=1741+12,1759=1741+18 and 1741=6*290+1
PrimePi/@Transpose[Select[Partition[Prime[Range[50000]],4,1], Differences[ #] == {6,6,6}&&Mod[#[[1]],6]==1&]][[1]] (* Harvey P. Dale, Nov 04 2015 *)
prime(54) = 251: 251, 257, 263, 269 are consecutive primes.
PrimePi/@Transpose[Select[Partition[Prime[Range[50000]],4,1],Differences[#]=={6, 6, 6}&&Mod[#[[1]],6]==5&]][[1]] (* Metin Sariyar, Sep 21 2019 *)
isok(n) = {my(p=prime(n), q, r, s); ((p % 6) == 5) && ((q=nextprime(p+1)) == p+6) && ((r=nextprime(q+1)) == p+12) && ((s=nextprime(r+1)) == p+18);} \\ Michel Marcus, Sep 20 2019
42 is in the sequence because 6*42-1=251, 6*43-1=257, 6*44-1=263, 6*45-1=296 are prime and there are no other primes in between, i.e., 6*42+1=253=11*23, 6*43+1=259=7*37, 6*44+1=265=5*53 are not primes.
Block[{nn = 50000, s}, s = Select[Prime@ Range@ PrimePi[6 (nn + 3) + 1], Divisible[(# - 1), 6] &]; Select[Range@ nn, And[AllTrue[#, PrimeQ], Count[s, q_ /; First[#] < q < Last@ #] == 0] &@ Map[6 # - 1 &, # + Range[0, 3]] &]] (* Michael De Vlieger, Dec 06 2017 *)
fQ[n_] := Block[{p = {6n -1, 6n +5, 6n +11, 6n +17}}, Union@ PrimeQ@ p == {True} && NextPrime[6n -1, 3] == 6n +17]; Select[Range@50000, fQ] (* Robert G. Wilson v, Dec 14 2017 *)
isok(n) = isprime(6*n-1) && isprime(6*n+5) && isprime(6*n+11) && isprime(6*n+17) && ((primepi(6*n+17) - primepi(6*n-1)) == 3); \\ Michel Marcus, Dec 11 2017
a, b, c, d = 2, 3, 5, 7; R = []
for p in primes(10**5):
if a % 6 + 1 == b - a == c - b == d - c == 6:
R.append((a+1)//6)
a, b, c, d = b, c, d, p
R # Peter Luschny, Jan 08 2018
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