A046118 -id:A046118 - cp's OEIS frontend

A046119 Middle member of a sexy prime triple: value of p+6 such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be).

13, 23, 37, 53, 73, 103, 107, 157, 173, 233, 263, 277, 353, 373, 563, 593, 613, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1283, 1297, 1367, 1433, 1453, 1493, 1613, 1663, 1753, 1783, 1873, 1907, 1993, 2137, 2287, 2383, 2417, 2683, 2693, 2713

Offset: 1

Eric W. Weisstein

nonn

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 middle member; e.g., a(4)=53 is the middle member of the sexy prime triple (47, 53, 59), but is also the third member of the sexy prime quadruple (41, 47, 53, 59). - Daniel Forgues, Aug 05 2009

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].

Cf. A023201, A046117.

Cf. A046118, A046120.

lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&!PrimeQ[p+18], AppendTo[lst, p+6]], {n, 7!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[400]],And@@PrimeQ[{#-6,#+6}]&&!PrimeQ[#+12]&] (* Harvey P. Dale, Nov 01 2011 *)

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

Definition edited by Daniel Forgues, Aug 12 2009

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

Eric W. Weisstein

nonn

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

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].

Cf. A023201, A046117.

Cf. A046118, A046119.

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 *)

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

A275681 Table read by rows: list of sexy prime triples (p, p+6, p+12) such that p+18 is composite.

Original entry on oeis.org

7, 13, 19, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 73, 79, 97, 103, 109, 101, 107, 113, 151, 157, 163, 167, 173, 179, 227, 233, 239, 257, 263, 269, 271, 277, 283, 347, 353, 359, 367, 373, 379, 557, 563, 569, 587, 593, 599, 607, 613, 619, 647, 653, 659, 727, 733, 739

Offset: 1

Arkadiusz Wesolowski, Aug 05 2016

			The table starts:
7, 13, 19;
17, 23, 29;
31, 37, 43;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Wikipedia, Sexy prime
Index entries for primes, gaps between

Cf. A023201 (sexy primes), A046118, A123082, A275682.

lst:=[]; for p in PrimesUpTo(727) do b:=p+6; if IsPrime(b) then c:=b+6; if IsPrime(c) and not IsPrime(c+6) then lst:=lst cat [p, b, c]; end if; end if; end for; lst;

N:= 10^4: # to get all entries <= N
Primes:= select(isprime,{seq(i,i=1..N+18,2)}):
S:= select(`<=`, Primes,N) intersect map(t -> t-6, Primes) intersect map(t -> t-12, Primes) minus map(t -> t-18, Primes):
map(t ->(t,t+6,t+12), sort(convert(S,list))); # Robert Israel, Aug 05 2016

Most[#]&/@Select[Table[n+{0,6,12,18},{n,Prime[Range[200]]}],PrimeQ[#] == {True,True,True,False}&]//Flatten (* Harvey P. Dale, Jan 19 2017 *)

a(3*n-2) = A046118(n).
a(3*n-1) = A046118(n)+6.
a(3*n) = A046118(n)+12.

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

Daniel Forgues, Aug 05 2009, Aug 12 2009

nonn

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).

Daniel Forgues, Table of n, a(n) for n=1..99982

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

Connor Zapfel, May 04 2017

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

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

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

Cf. A023201, A046118, A046119, A046120.

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 *)

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

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

A275686 Difference between the smallest 10^n-digit member of a sexy prime triple and 10^(10^n - 1).

Original entry on oeis.org

6, 427, 264607, 4975694077

Offset: 0

Arkadiusz Wesolowski, Aug 05 2016

Cf. A046118.

A297847 Sexiness of p = prime(n): number of iterations of the function f(x) = x + 6 that leave p prime.

Original entry on oeis.org

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

Offset: 1

Felix Fröhlich, Jan 07 2018

nonn

a(n) > 0 iff p is a term of A023201.
a(n) = 0 iff p is a term of A140555.
a(n) = 2 iff p is a term of A046118.
a(n) > 2 iff p is a term of A023271.
a(n) < 4 except for n = 3. Proof: The last digits of the numbers in the progression repeat 1, 7, 3, 9, 5, 1, 7, 3, 9, 5, ..., so a(n) is at most 4, which only happens for p = 5, since A007652(n) = 5 only for n = 3.

			For n = 13: prime(13) = 41 and 41 remains prime through exactly 3 iterations of f(x) = x + 6, since 47, 53 and 59 are prime, but 65 is composite, so a(13) = 3.

Wikipedia, Sexy prime

Cf. A023201, A023271, A046118, A140555.

Array[-2 + Length@ NestWhileList[# + 6 &, Prime@ #, PrimeQ] &, 105] (* Michael De Vlieger, Jan 11 2018 *)

a(n) = my(p=prime(n), x=p, i=0); while(1, x=x+6; if(!ispseudoprime(x), return(i), i++))

A358572 Smallest prime p in a sexy prime triple such that (p-3)/2 is also the smallest prime in a sexy prime triple (A023241).

Original entry on oeis.org

17, 97, 1117, 1217, 2897, 130337, 188857, 207997, 221197, 324517, 610817, 900577, 1090877, 1452317, 1719857, 1785097, 2902477, 3069917, 3246317, 4095097, 4536517, 4977097, 5153537, 5517637, 5745557, 6399677, 7168277, 7351957, 7588697, 7661077, 8651537, 8828497, 9153337

Offset: 1

Lamine Ngom, Nov 23 2022

nonn

Also numbers m such that m-4, m-1, m+5, m+8, m+11 and m+20 cannot be represented as x*y + x + y, with x >= y > 1 (A254636).
Subsequence of A358571.
Number of terms < 10^k: 0, 2, 2, 5, 5, 12, 34, 150, 655, ...
All terms p and (p-3)/2 have a final decimal digit of 7. This follows from considering possibilities modulo 10 and implies p + 18 and (p-3)/2 + 18 are divisible by 5 and hence composite. Both p and (p-3)/2 are therefore also terms of A046118. - Andrew Howroyd, Dec 31 2022

			97 is the smallest prime in the sexy prime triple (97, 103, 109), and the triple (47 = (97 - 3)/2, 47 + 6, 47 + 12) forms another sexy prime triple. Hence 97 is in the sequence.

Cf. A023201, A023241, A046118, A255361, A254636, A256386, A358571.

Select[Prime[Range[700000]], AllTrue[Join[# + {6,12}, (#-3)/2 + {0, 6, 12}], PrimeQ] &] (* Amiram Eldar, Nov 23 2022 *)

istriple(p)={isprime(p) && isprime(p+6) && isprime(p+12)}
isok(p)={istriple(p) && istriple((p-3)/2)}
{ forprime(p=1,10^7,if(isok(p), print1(p, ", "))) } \\ Andrew Howroyd, Dec 30 2022

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A046119 Middle member of a sexy prime triple: value of p+6 such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be).

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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.

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A275681 Table read by rows: list of sexy prime triples (p, p+6, p+12) such that p+18 is composite.

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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.

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A286217 Product of the n-th sexy prime triple.

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

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Maple

A275686 Difference between the smallest 10^n-digit member of a sexy prime triple and 10^(10^n - 1).

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A297847 Sexiness of p = prime(n): number of iterations of the function f(x) = x + 6 that leave p prime.

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A358572 Smallest prime p in a sexy prime triple such that (p-3)/2 is also the smallest prime in a sexy prime triple (A023241).

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