A023271 -id:A023271 - cp's OEIS frontend

A227286 First primes of arithmetic progressions of 13 primes each with the common difference 30030.

14933623, 2085471361, 132420258931, 185041386139, 682539280751, 834172298383, 834172328413, 856378247603, 856378277633, 888867525577, 931115864233, 1059709587163, 1345030977911, 1360910561113, 1578280523803, 1973348047529, 1988253536611, 2083502941613

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

The minimal possible difference in an arithmetic progression of k primes is conjectured to be k# = A034386(k) for all k > 7. 13# = 30030.

			p = 2085471361 then the AP-13 is {2085471361, 2085501391, 2085531421, 2085561451, 2085591481, 2085621511, 2085651541, 2085681571, 2085711601, 2085741631, 2085771661, 2085801691, 2085831721} with the difference 13# = 2*3*5*7*11*13 = 30030.

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227283, A227284, A227285.

Clear[p]; d = 30030; ap13p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d, p + 8*d, p + 9*d, p + 10*d, p + 11*d, p + 12*d}] == {True, True, True, True, True, True, True, True, True, True, True, True, True}, AppendTo[ap13p, p]], {p, 3, 41*10^9, 2}]; ap13p

More terms from Jens Kruse Andersen, Jun 27 2014

A275682 Table read by rows: list of sexy prime quadruples (p, p+6, p+12, p+18) such that p+24 is composite.

Original entry on oeis.org

11, 17, 23, 29, 41, 47, 53, 59, 61, 67, 73, 79, 251, 257, 263, 269, 601, 607, 613, 619, 641, 647, 653, 659, 1091, 1097, 1103, 1109, 1481, 1487, 1493, 1499, 1601, 1607, 1613, 1619, 1741, 1747, 1753, 1759, 1861, 1867, 1873, 1879, 2371, 2377, 2383, 2389

Offset: 1

Arkadiusz Wesolowski, Aug 05 2016

(5, 11, 17, 23, 29) is a sexy prime 5-tuple and this is the only sexy prime 5-tuple.

Essentially same as A123083.

			The table starts:
11, 17, 23, 29;
41, 47, 53, 59;
61, 67, 73, 79;
...

Vincenzo Librandi, Table of n, a(n) for n = 1..2100
Wikipedia, Sexy prime
Index entries for primes, gaps between

Cf. A023201, A023271, A123083, A275681.

lst:=[]; for p in PrimesInInterval(7, 2371) do b:=p+6; if IsPrime(b) then c:=b+6; if IsPrime(c) then d:=c+6; if IsPrime(d) then lst:=lst cat [p, b, c, d]; end if; end if; end if; end for; lst;

Most[#]&/@Select[Table[n + {0, 6, 12, 18, 24}, {n, Prime[Range[200]]}], PrimeQ[#]=={True, True, True, True, False}&]//Flatten (* Vincenzo Librandi, Jun 09 2017 *)
#+{0,6,12,18}&/@Select[Prime[Range[400]],AllTrue[#+{6,12,18},PrimeQ] && CompositeQ[#+24]&]//Flatten (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 29 2019 *)

a(n) = A123083(n+4).

A377318 Numbers k such that prime(k), prime(k)+6, prime(k)+12, and prime(k)+18 are primes.

Original entry on oeis.org

3, 5, 13, 18, 54, 110, 116, 182, 234, 252, 271, 284, 351, 387, 464, 541, 551, 682, 709, 717, 741, 821, 829, 1171, 1417, 1448, 1510, 1594, 1711, 1726, 1842, 1853, 2009, 2086, 2209, 2297, 2408, 2600, 2680, 2876, 2924, 2930, 3253, 3303, 3437, 3977, 4384, 4431

Offset: 1

Kritsada Moomuang, Oct 24 2024

nonn

			5 is in this sequence because: prime(5) = 11 and 11+6=17, 11+12=23, and 11+18=29 are all primes.

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

Cf. A000040, A000720, A023271.

Subsequence of A377317.

Select[Range[1, PrimePi[50000]], PrimeQ[Prime[#] + 6] && PrimeQ[Prime[#] + 12] && PrimeQ[Prime[#] + 18] &]
Select[Range[4500],AllTrue[Prime[#]+{0,6,12,18},PrimeQ]&] (* Harvey P. Dale, Jun 17 2025 *)

for(k=1, primepi(50000), p = prime(k); if(isprime(p+6) && isprime(p+12) && isprime(p+18), print(k)))

a(n) = pi(A023271(n)).

A092519 Smallest prime a(n) such that a(n)+6n, a(n)+12n and a(n)+18n are also primes.

Original entry on oeis.org

5, 5, 5, 59, 7, 31, 5, 5, 5, 11, 31, 7, 23, 5, 11, 5, 7, 23, 79, 29, 5, 5, 73, 29, 7, 41, 107, 43, 19, 59, 11, 37, 13, 79, 13, 11, 17, 43, 359, 23, 31, 5, 53, 19, 47, 181, 137, 23, 59, 7, 491, 127, 283, 179, 23, 11, 7, 5, 89, 7, 461, 7, 53, 139, 31, 5, 7, 13

Offset: 1

Ray G. Opao, Apr 06 2004

nonn

			a(5) = 7: 7+6*5 = 7+30 = 37, 7+12*5 = 7+60 = 67, and 7+18*5 = 7+90 = 97 are all prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

a(1) starts sequence A023271.

prs := [NthPrime(i) : i in [1..200]]; result := []; for n in [1..100] do  for p in prs do if forall{k : k in [1..3] | IsPrime(p + 6*n*k)} then  Append(~result, p); break; end if; end for; end for; result; // Vincenzo Librandi, Jul 26 2025

Module[{nn=100,prs},prs=Prime[Range[nn]];Table[SelectFirst[prs, AllTrue[ #+6*Range[3]*n,PrimeQ]&],{n,nn/2}]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 22 2018 *)

A160264 Least prime of a 6-tuplet that contains both a prime quadruple and a sexy prime quadruple.

Original entry on oeis.org

11, 1481, 1861, 5641, 88801, 165701, 266671, 284731, 326141, 402751, 626611, 661091, 855721, 959461, 1022501, 1068241, 1068701, 1118851, 1146781, 1155601, 1246361, 1461401, 1573921, 1830331, 1917731, 2674531, 2683771, 3058871

Offset: 1

Ki Punches, May 05 2009

nonn

Sequence is probably infinite.

			6-tuplet 5641 5647 5651 5653 5657 5659, contains sexy prime quadruple 5641 5647 5653 5659, and prime quadruple 5651 5653 5657 5659.

Harvey P. Dale, Table of n, a(n) for n = 1..165

Cf. A023271, A007530.

stQ[n_]:=Module[{ss4=Subsets[n,{4}]},AnyTrue[ss4,Differences[#]=={6,6,6}&] && AnyTrue[ss4,Differences[#]=={2,4,2}&]]; Select[Partition[Prime[ Range[ 221000]],6,1],stQ][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 24 2020 *)

Extended by Ray Chandler, May 23 2009

A163857 Number of sexy prime quadruples (p, p+6, p+12, p+18), with p <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Daniel Forgues, Aug 05 2009

nonn

There are 2 sexy prime quadruples classes, (-1, -1, -1, -1) (mod 6) and (+1, +1, +1, +1) (mod 6). They should asymptotically have the same number of quadruples, if there is an infinity of such quadruples, although with a Chebyshev bias expected against the quadratic residue class quadruples (+1, +1, +1, +1) (mod 6), which doesn't affect the asymptotic result. This sequence counts both classes.
Also the sexy prime quadruples of class (-1, -1, -1, -1) (mod 6) are (11, 17, 23, 29) (mod 30) while the sexy prime quadruples of class (+1, +1, +1, +1) (mod 6) are (1, 7, 13, 19) (mod 30).
Except for (5, 11, 17, 23, 29), there is no sexy prime quintuples (p, p+6, p+12, p+18, p+24) since one of the members is then divisible by 5.

Daniel Forgues, Table of n, a(n) for n=1..99982
Eric Weisstein's World of Mathematics, Prime Constellation

A023271 First member of a sexy prime quadruple: value of p where (p, p+6, p+12, p+18) are all prime.
A046122 Second member of a sexy prime quadruple: value of p+6 where (p, p+6, p+12, p+18) are all prime.
A046123 Third member of a sexy prime quadruple: value of p+12 where (p, p+6, p+12, p+18) are all prime.
A046124 Last member of a sexy prime quadruple: value of p+18 where (p, p+6, p+12, p+18) are all prime.

A357909 Primes p such that p+6, p+12, p+18, 4p+37, 4p+43, 4p+49 and 4p+55 are also all primes.

Original entry on oeis.org

408211, 6375751, 6433741, 6718471, 19134931, 25280791, 63908851, 67078801, 152418151, 159268561, 217697911, 236220991, 237943591, 334030981, 363246211, 392644921, 406249171, 410652031, 428032441, 476660281, 478441291, 502777111, 552727711, 552855001, 554201731, 693654721, 816050071, 877207141

Offset: 1

J. M. Bergot and Robert Israel, Nov 09 2022

nonn

Start of a "sexy" prime quadruple (in the sense of A023271) such that 1 + the sum of the quadruple is the start of another "sexy" prime quadruple.

All terms == 1 (mod 30).

			a(1) = 408211 is a term because 408211, 408211+6 = 408217, 408211+12 = 408223, 408211+18 = 408229 are primes (a "sexy" prime quadruple), the sum of this quadruple is 4*408211+36 = 1632880, and 1632880+1 = 1632881, 1632880+7 = 1632887, 1632880+13 = 1632893, 1632880+19 = 1632899 is another "sexy" prime quadruple.

Robert Israel, Table of n, a(n) for n = 1..115

Cf. A023271.

Res:= NULL: count:= 0:
for p from 1 by 30 while count < 40 do
  if isprime(p) and isprime(p+6) and isprime(p+12) and isprime(p+18)
  and isprime(4*p+37) and isprime(4*p+43) and isprime(4*p+49) and isprime(4*p+55)
then Res:= Res, p; count:= count+1
fi
od:
Res;

A358322 Interlopers in sexy prime quadruples.

Original entry on oeis.org

7, 13, 19, 43, 71, 617, 643, 1093, 1483, 1489, 1609, 1871, 1877, 2381, 2687, 3919, 4003, 5441, 5651, 5657, 9463, 11831, 12109, 14629, 20357, 21491, 24107, 26683, 26713, 32059, 37571, 41957, 42407, 44533, 50591, 55217, 65717, 68899, 70001, 79813, 87557, 88811, 88817, 103993, 110923, 112573, 122029

Offset: 1

J. M. Bergot and Robert Israel, Nov 09 2022

nonn

Primes q !== p (mod 6) such that p < q < p+18, where (p, p+6, p+12, p+18) is a "sexy" prime quadruple, i.e., p is in A023271.

			a(5) = 71 is a term because it is a prime !== 61 (mod 6) with 61 < 71 < 79, where (61, 67, 73, 79) is a sexy prime quadruple.

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

Cf. A023271.

Res:= 7: count:= 1:
for p from 11 by 10 while count < 100 do
  if andmap(isprime, [p, p+6, p+12, p+18]) then
    R:= select(isprime, [p+2, p+8, p+10, p+16]);
    count:= count + nops(R);
    Res:= Res, op(R);
  fi
od:
Res;

A358343 Primes p such that p + 6, p + 12, p + 18, (p+4)/5, (p+4)/5 + 6, (p+4)/5 + 12 and (p+4)/5 + 18 are also prime.

Original entry on oeis.org

213724201, 336987901, 791091901, 1940820901, 2454494551, 2525191051, 2675901751, 3490984201, 3571597951, 3702692551, 4045565851, 4531570951, 5698472701, 5928161251, 5953041001, 6589503751, 7073836201, 7360771801, 7811308951, 8282895451, 10242069451, 11049315751, 12392801251, 13062696001

Offset: 1

J. M. Bergot and Robert Israel, Nov 10 2022

nonn

Terms p of A023271 such that (p+4)/5 is also in A023271.

All terms == 901 (mod 1050).

			a(3) = 791091901 is a term because p = 791091901, p + 6 = 791091907, p + 12 = 791091913, p + 18 = 791091919, (p+4)/5 = 158218381, (p+4)/5 + 6 = 158218387, (p+4)/5 + 12 = 158218393, and (p+4)/5 + 18 = 158218399 are all prime.

Robert Israel, Table of n, a(n) for n = 1..213

Cf. A023271.

filter:= p -> andmap(isprime, [p, p+6, p+12, p+18, (p+4)/5, (p+4)/5 + 6,
(p+4)/5 + 12, (p+4)/5 + 18]):
select(filter, [seq(p, p = 901 .. 2*10^10, 1050)]);

A372042 Monogamously Faithful Primes (primes that are sexy primes with only one other prime in their pair).

Original entry on oeis.org

83, 89, 131, 137, 191, 193, 197, 199, 223, 229, 307, 311, 313, 317, 331, 337, 383, 389, 433, 439, 443, 449, 457, 461, 463, 467, 503, 509, 541, 547, 571, 577, 677, 683, 751, 757, 821, 823, 827, 829, 853, 857, 859, 863, 877, 881, 883, 887, 991, 997, 1013, 1019, 1033, 1039, 1063, 1069, 1087

Offset: 0

Ryan Stoler, Apr 17 2024

nonn

These are all the numbers found in A136207 but not found in A046118, A046119, A046120, A023271, A046122, A046123, or A046124, i.e., members of a sexy prime pair but not members of sexy prime triplets, quadruplets, ...

			83 and 89 are "sexy" with each other, because they differ by 6. They are monogamously faithful, because neither is sexy with any other number.
71 is not "sexy" because it is not in A136207.
67 is "sexy" with both 61 and 73. Therefore, it is not monogamously faithful, since it has multiple numbers that it is sexy with.
43 is "sexy" only with 37. But it is not monogamously faithful, even though it isn't sexy with another number, because 37 is also "sexy" with 31, therefore "cheating" on 43 with 31.

Cf. A136207, A046117, A046118, A046119, A046120, A023271, A046122, A046123, A046124.

isA372042 := proc(n)
    if isprime(n) then
        if isprime(n+6) then
            if not isprime(n-6) and not isprime(n+12) then
                true;
            else
                false;
            end if;
        elif isprime(n-6) then
            if not isprime(n+6) and not isprime(n-12) then
                true;
            else
                false;
            end if;
        else
            false ;
        end if;
    else
        false ;
    end if;
end proc:
A372042 := proc(n)
    option remember;
    local a;
    if n = 1 then
        83 ;
    else
        a := nextprime(procname(n-1)) ;
        while true do
            if isA372042(a) then
                return a;
            else
                a := nextprime(a) ;
            end if;
        end do:
    end if;
end proc:
seq(A372042(n),n=1..80) ; # R. J. Mathar, Jun 10 2024

cp's OEIS Frontend

A227286 First primes of arithmetic progressions of 13 primes each with the common difference 30030.

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A275682 Table read by rows: list of sexy prime quadruples (p, p+6, p+12, p+18) such that p+24 is composite.

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Magma

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A377318 Numbers k such that prime(k), prime(k)+6, prime(k)+12, and prime(k)+18 are primes.

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A092519 Smallest prime a(n) such that a(n)+6n, a(n)+12n and a(n)+18n are also primes.

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Magma

Mathematica

A160264 Least prime of a 6-tuplet that contains both a prime quadruple and a sexy prime quadruple.

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A163857 Number of sexy prime quadruples (p, p+6, p+12, p+18), with p <= n.

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A357909 Primes p such that p+6, p+12, p+18, 4*p+37, 4*p+43, 4*p+49 and 4*p+55 are also all primes.

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Maple

A358322 Interlopers in sexy prime quadruples.

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A357909 Primes p such that p+6, p+12, p+18, 4p+37, 4p+43, 4p+49 and 4p+55 are also all primes.