cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A046118 Smallest member of a sexy prime triple: value of p such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be).

Original entry on oeis.org

7, 17, 31, 47, 67, 97, 101, 151, 167, 227, 257, 271, 347, 367, 557, 587, 607, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1277, 1291, 1361, 1427, 1447, 1487, 1607, 1657, 1747, 1777, 1867, 1901, 1987, 2131, 2281, 2377, 2411, 2677, 2687, 2707, 2791, 2897, 2957
Offset: 1

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Author

Eric W. Weisstein

Keywords

Comments

p-6 will be prime if the prime triple contains the last 3 primes of a sexy prime quadruple.
If a sexy prime triple happens to include the last 3 members of a sexy prime quadruple, this sequence will contain the sexy prime triple's smallest member; e.g., a(4)=47 is the smallest member of the sexy prime triple (47, 53, 59), but is also the second member of the sexy prime quadruple (41, 47, 53, 59). - Daniel Forgues, Aug 05 2009

Links

Crossrefs

Cf. A023201, A046117.
Cf. A046119, A046120.

Programs

  • Magma
    [p: p in PrimesUpTo(5000) | not IsPrime(p+18) and IsPrime(p+6) and IsPrime(p+12)]; // Vincenzo Librandi, Sep 07 2017
  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&!PrimeQ[p+18], AppendTo[lst, p]], {n, 7!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[500]],AllTrue[#+{6,12},PrimeQ]&&CompositeQ[#+18]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 11 2019 *)
  • PARI
    lista(nn) = forprime(p=3, nn, if (isprime(p+6) && isprime(p+12) && !isprime(p+18), print1(p, ", "));); \\ Michel Marcus, Jan 06 2015

Extensions

Definition edited by Daniel Forgues, Aug 12 2009
More terms from Eric M. Schmidt, Sep 07 2017

A046120 Largest member of a sexy prime triple; value of p+12 where p, p+6 and p+12 are all prime, but p+18 is not.

Original entry on oeis.org

19, 29, 43, 59, 79, 109, 113, 163, 179, 239, 269, 283, 359, 379, 569, 599, 619, 659, 739, 953, 983, 1109, 1129, 1193, 1229, 1289, 1303, 1373, 1439, 1459, 1499, 1619, 1669, 1759, 1789, 1879, 1913, 1999, 2143, 2293, 2389, 2423, 2689, 2699, 2719
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

If a sexy prime triple happens to include the last 3 members of a sexy prime quadruple, this sequence will contain the sexy prime triple's largest member; e.g., a(4)=59 is the largest member of the sexy prime triple (47, 53, 59), but is the fourth member of the sexy prime quadruple (41, 47, 53, 59). - Daniel Forgues, Aug 05 2009

Links

  • Harvey P. Dale, Table of n, a(n) for n = 1..1000
  • Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Crossrefs

Cf. A023201, A046117.
Cf. A046118, A046119.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&!PrimeQ[p+18], AppendTo[lst, p+12]], {n, 7!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
#+12&/@Select[Prime[Range[400]],PrimeQ[#+{6,12,18}]=={True,True,False}&] (* Harvey P. Dale, Dec 08 2012 *)

Formula

a(n) = A046118(n)+12 and a(n) = A046119(n)+6. - Michel Marcus, Jan 06 2015

A163858 Number of sexy prime triples (p, p+6, p+12) where p+18 is not prime (although p-6 might be), with p <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7
Offset: 1

Views

Author

Daniel Forgues, Aug 05 2009, Aug 12 2009

Keywords

Comments

p-6 will be prime if the prime triple contains the last 3 primes of a sexy prime quadruple.
There are two sexy prime triples classes, (-1, -1, -1) (mod 6) and (+1, +1, +1) (mod 6). They should asymptotically have the same number of triples, if there is an infinity of such triples, although with a Chebyshev bias expected against the quadratic residue class triples (+1, +1, +1) (mod 6), which doesn't affect the asymptotic result. This sequence counts both classes.
Also the sexy prime triples of class (-1, -1, -1) (mod 6) fall within (11, 17, 23, 29) (mod 30) while the sexy prime triples of class (+1, +1, +1) (mod 6) fall within (1, 7, 13, 19) (mod 30).

Links

Crossrefs

A046118 Smallest member of a sexy prime triple: value of p where (p, p+6, p+12) are all prime but p+18 is not (although p-6 might be.)
A046119 Middle member of a sexy prime triple: value of p+6 where (p, p+6, p+12) are all prime but p+18 is not (although p-6 might be.)
A046120 Largest member of a sexy prime triple, value of p+12 where (p, p+6, p+12) are all prime but p+18 is not (although p-6 might be.)

A286217 Product of the n-th sexy prime triple.

Original entry on oeis.org

1729, 11339, 49321, 146969, 386389, 1089019, 1221191, 3864241, 5171489, 12640949, 18181979, 21243961, 43974269, 51881689, 178433279, 208506509, 230324329, 278421569, 393806449, 849244031, 932539661, 1341880019, 1416207439, 1672403471, 1829232539, 2111885999
Offset: 1

Views

Author

Connor Zapfel, May 04 2017

Keywords

Comments

A sexy prime triple is such that p, p+6, and p+12 are primes but p+18 is not a prime. - Harvey P. Dale, Oct 13 2024

Examples

			The first sexy prime triple is (7, 13, 19) so a(1) = 7*13*19 = 1729.

Links

Crossrefs

Cf. A023201, A046118, A046119, A046120.

Programs

  • Mathematica
    Select[Prime@ Range@ 500, PrimeQ[# + {6, 12, 18}] == {True, True, False} &] // # (#+6) (#+12) & (* Giovanni Resta, May 05 2017 *)
Times@@Take[#,3]&/@(Select[Table[p+{0,6,12,18},{p,Prime[Range[250]]}],Boole[PrimeQ[#]]=={1,1,1,0}&]) (* Harvey P. Dale, Oct 13 2024 *)

Formula

a(n) = s(n)*(s(n)+6)*(s(n)+12), where s = A046118.
a(n) = A046118(n) * A046119(n) * A046120(n).

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

Views

Author

Ryan Stoler, Apr 17 2024

Keywords

Comments

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, ...

Examples

			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.

Crossrefs

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

Programs

  • Maple
    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
Showing 1-5 of 5 results.

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