A046117 -id:A046117 - cp's OEIS frontend

A023201 Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)

5, 7, 11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 131, 151, 157, 167, 173, 191, 193, 223, 227, 233, 251, 257, 263, 271, 277, 307, 311, 331, 347, 353, 367, 373, 383, 433, 443, 457, 461, 503, 541, 557, 563, 571, 587, 593, 601, 607, 613, 641, 647

Offset: 1

David W. Wilson

T. D. Noe, Table of n, a(n) for n = 1..10000
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
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].
Wikipedia, Sexy prime.

A031924 (primes starting a gap of 6) and A007529 together give this (A023201).

Cf. A046117 (a(n)+6), A087695 (a(n)+3), A098428, A000040, A010051, A006489 (subsequence).

a023201 n = a023201_list !! (n-1)
a023201_list = filter ((== 1) . a010051 . (+ 6)) a000040_list
-- Reinhard Zumkeller, Feb 25 2013

[n: n in [0..40000] | IsPrime(n) and IsPrime(n+6)]; // Vincenzo Librandi, Aug 04 2010

A023201 := proc(n)
    option remember;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+2 by 2 do
            if isprime(a) and isprime(a+6) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 28 2013

Select[Range[10^2], PrimeQ[ # ]&&PrimeQ[ #+6] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[120]],PrimeQ[#+6]&] (* Harvey P. Dale, Mar 20 2018 *)

is(n)=isprime(n+6)&&isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From M. F. Hasler, Jan 02 2020: (Start)
a(n) = A046117(n) - 6 = A087695(n) - 3.
A023201 = { p = A000040(k) | A000040(k+1) = p+6 or A000040(k+2) = p+6 } = A031924 U A007529. (End)

A023271 Primes p such that p, p+6, p+12, p+18 are all primes.

Original entry on oeis.org

5, 11, 41, 61, 251, 601, 641, 1091, 1481, 1601, 1741, 1861, 2371, 2671, 3301, 3911, 4001, 5101, 5381, 5431, 5641, 6311, 6361, 9461, 11821, 12101, 12641, 13451, 14621, 14741, 15791, 15901, 17471, 18211, 19471, 20341, 21481, 23321, 24091, 26171, 26681

Offset: 1

David W. Wilson

nonn

Smallest member of a "sexy" prime quadruple.
For n > 1, a(n) ends in 1. - Robert Israel, Jul 16 2015
The only sexy prime quintuple corresponding to (p, p+6, p+12, p+18, p+24) starts with a(1) = 5, so this quintuple is (5, 11, 17, 23, 29) (see Wikipedia link and A206039). - Bernard Schott, Mar 10 2023

Matt C. Anderson and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..100 from Matt C. Anderson)
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]
Wikipedia, Sexy prime.

Cf. A023201, A046117, A046122, A046123, A046124, A206039.

[p: p in PrimesInInterval(2, 1000000) | forall{i: i in [ 6, 12, 18] | IsPrime(p+i)}]; // Vincenzo Librandi, Jul 15 2015

for a to 2*10^5 do
if `and`(isprime(a), isprime(a+6), isprime(a+12), isprime(a+18))
then print(a);
end if;
end do;
# code produces 109 primes
# Matt C. Anderson, Jul 15 2015

Select[Prime[Range[1000]], PrimeQ[# + 6] && PrimeQ[# + 12] && PrimeQ[# + 18] &] (* Vincenzo Librandi, Jul 15 2015 *)
(* The following program uses the AllTrue function from Mathematica version 10 *) Select[Prime[Range[3000]], AllTrue[# + {6, 12, 18}, PrimeQ] &] (* Harvey P. Dale, Jun 06 2017 *)

main(size)=my(v=vector(size),i,r=1,p);for(i=1,size,while(1,p=prime(r);if(isprime(p+6)&&isprime(p+12)&&isprime(p+18),v[i]=p;r++;break,r++))); v \\ Anders Hellström, Jul 16 2015

Edited by N. J. A. Sloane, Aug 04 2009 following a suggestion from Daniel Forgues

A087695 Numbers n such that n + 3 and n - 3 are both prime.

Original entry on oeis.org

8, 10, 14, 16, 20, 26, 34, 40, 44, 50, 56, 64, 70, 76, 86, 100, 104, 106, 110, 134, 154, 160, 170, 176, 194, 196, 226, 230, 236, 254, 260, 266, 274, 280, 310, 314, 334, 350, 356, 370, 376, 386, 436, 446, 460, 464, 506, 544, 560, 566, 574, 590, 596

Offset: 1

Zak Seidov, Sep 27 2003

A010051(a(n)-3) * A010051(a(n)+3) = 1. - Reinhard Zumkeller, Nov 17 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A014574, A087678-A087683, A087696, A087697, A088763, A046117, A010051.

a087695 n = a087695_list !! (n-1)
a087695_list = filter
   (\x -> a010051' (x - 3) == 1 && a010051' (x + 3) == 1) [2, 4 ..]
-- Reinhard Zumkeller, Nov 17 2015

ZL:=[]:for p from 1 to 600 do if (isprime(p) and isprime(p+6) ) then ZL:=[op(ZL),(p+(p+6))/2]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

lst={};Do[If[PrimeQ[n-3]&&PrimeQ[n+3], AppendTo[lst, n]], {n, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
Select[Range[600],AllTrue[#+{3,-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 06 2015 *)

p=2; q=3; forprime(r=5,1e3, if(q-p<7 && (q-p==6 || r-p==6), print1(p+3", ")); p=q; q=r) \\ Charles R Greathouse IV, May 22 2018

a(n) = A046117(n) - 3.

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

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 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

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
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. A046119, A046120.

[p: p in PrimesUpTo(5000) | not IsPrime(p+18) and IsPrime(p+6) and IsPrime(p+12)]; // Vincenzo Librandi, Sep 07 2017

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 *)

lista(nn) = forprime(p=3, nn, if (isprime(p+6) && isprime(p+12) && !isprime(p+18), print1(p, ", "));); \\ Michel Marcus, Jan 06 2015

Definition edited by Daniel Forgues, Aug 12 2009

More terms from Eric M. Schmidt, Sep 07 2017

A098412 Greatest members p of prime triples (p-6, p-4, p).

Original entry on oeis.org

11, 17, 23, 47, 107, 113, 197, 233, 317, 353, 467, 647, 827, 863, 887, 1097, 1283, 1307, 1433, 1487, 1493, 1613, 1877, 2003, 2087, 2243, 2273, 2663, 2693, 3257, 3467, 3533, 3677, 3923, 4007, 4133, 4523, 4643, 4793, 4937, 4973, 5237, 5483, 5507, 5657, 6203

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Subsequence of A046117; a(n) = A073648(n) + 4 = A022004(n) + 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098413, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-4)]; // Vincenzo Librandi, Dec 26 2010

K:=10^7: # to get all terms <= K.
for n from 1 by 2 to K do; if isprime(n-6) and isprime(n-4) and isprime(n) then print(n) else fi; od;  # Muniru A Asiru, Aug 06 2017

Select[Table[Prime[n], {n, 1000}], PrimeQ[# - 4] && PrimeQ[# - 6] &] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&][[All,3]] (* Harvey P. Dale, Sep 23 2017 *)

A111192 Product of the n-th sexy prime pair.

Original entry on oeis.org

55, 91, 187, 247, 391, 667, 1147, 1591, 1927, 2491, 3127, 4087, 4891, 5767, 7387, 9991, 10807, 11227, 12091, 17947, 23707, 25591, 28891, 30967, 37627, 38407, 51067, 52891, 55687, 64507, 67591, 70747, 75067, 78391, 96091, 98587, 111547, 122491, 126727, 136891

Offset: 1

Shawn M Moore (sartak(AT)gmail.com), Oct 23 2005

nonn

Semiprime of the form 4*m^2-9 = (2*m-3)*(2*m+3). - Vincenzo Librandi, Jan 26 2016

			a(2)=91 because the second sexy prime pair is (7, 13) and 7*13=91.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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, A104229.

Cf. A037074, A143206, A195118; intersection of A143205 and A001358.

a111192 n = a111192_list !! (n-1)
a111192_list = f a000040_list where
   f (p:ps@(q:r:_)) | q - p == 6 = (p*q) : f ps
                    | r - p == 6 = (p*r) : f ps
                    | otherwise  = f ps
-- Reinhard Zumkeller, Sep 13 2011

IsSemiprime:=func; [s: n in [1..300] | IsSemiprime(s) where s is 4*n^2-9]; // Vincenzo Librandi, Jan 26 2016

#(#+6)&/@Select[Prime[Range[100]], PrimeQ[#+6]&] (* Harvey P. Dale, Dec 17 2010 *)
(* For checking large numbers, the following code is better. For instance, we could use the fQ function to determine that 229031718473564142083 is not in this sequence. *) fQ[n_] := Block[{fi = FactorInteger[n]}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {6}]; Select[ Range[125000], fQ] (* Robert G. Wilson v, Feb 08 2012 *)
Select[Table[4 n^2 - 9, {n, 300}], PrimeOmega[#] == 2 &] (* Vincenzo Librandi, Jan 26 2016 *)

a(n) = A023201(n) * A046117(n). - Reinhard Zumkeller, Sep 13 2011

A098413 Greatest members p of prime triples (p-6, p-2, p).

Original entry on oeis.org

13, 19, 43, 73, 103, 109, 199, 229, 283, 313, 463, 619, 829, 859, 883, 1093, 1303, 1429, 1453, 1489, 1669, 1699, 1789, 1873, 1879, 1999, 2089, 2143, 2383, 2689, 2713, 2803, 3169, 3259, 3463, 3469, 3853, 4159, 4519, 4789, 5233, 5419, 5443, 5653, 5659, 5743

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Subsequence of A046117; a(n) = A073649(n) + 2 = A022005(n) + 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098412, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-2)]; // Vincenzo Librandi, Dec 26 2010

Transpose[Select[Partition[Prime[Range[800]],3,1],Differences[#] == {4,2}&]][[3]] (* Harvey P. Dale, Aug 21 2013 *)

A103206 Concatenations of pairs of primes that differ by 6.

Original entry on oeis.org

511, 713, 1117, 1319, 1723, 2329, 3137, 3743, 4147, 4753, 5359, 6167, 6773, 7379, 8389, 97103, 101107, 103109, 107113, 131137, 151157, 157163, 167173, 173179, 191197, 193199, 223229

Offset: 0

Jonathan Vos Post, Mar 19 2005

Prime in this sequence: a(3) = 1117, a(4) = 1319, a(5) = 1723, a(7) = 3137, a(15) = 8389, a(16) = 97103, a(17) = 101107, a(21) = 151157, a(22) = 157163, a(23) = 167173, a(27) = 223229. Semiprimes in this sequence: a(1) = 511 = 7 x 73, a(2) = 713 = 23 * 31, a(6) = 2329 = 17 * 137, a(8) = 3743 = 19 * 197, a(11) = 5359 = 23 * 233, a(12) = 6167 = 7 * 881, a(13) = 6773 = 13 * 521, a(14) = 7379 = 47 x 157, a(18) = 103109 = 23 * 4483, a(20) = 131137 = 71 * 1847, a(26) = 193199 = 43 * 4493, ... Note that a(9) = 4147 = 11 * 13 * 29 and a(19) = 107113 = 43 * 47 * 53 are the products of three primes with the same number of digits.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Sexy Primes.
T. Trotter, "Sexy Primes."

Cf. A023201, A046117.

Cf. A000040, A103195.

FromDigits[Flatten[IntegerDigits/@{#,#+6}]]&/@Select[Prime[Range[50]], PrimeQ[#+6]&] (* Harvey P. Dale, Jun 24 2015 *)

a(n) = A023201(n) concatenated with A023201(n)+6.

A160370 Smaller member p of a pair (p,p+6) of consecutive primes in different centuries.

Original entry on oeis.org

1097, 2897, 3797, 4597, 5297, 5897, 9397, 11497, 11897, 12197, 12497, 12697, 15797, 16097, 18797, 19597, 21997, 24097, 24197, 28597, 28697, 29297, 30097, 30197, 30697, 32497, 35597, 36997, 39097, 40897, 41597, 41897, 42397, 45497, 47297

Offset: 1

Ki Punches, May 11 2009

nonn

Note that the smaller member of a pair of sexy primes with the same constraint on centuries defines a different sequence, since members of a sexy prime pair do not need to be *consecutive* primes.
The larger member in the pair is obtained by adding 6 to an entry.
Every a(n)+3 is a multiple of 100 such that neither a(n)+2 nor a(n)+4 are primes. It appears that every integer occurs as the difference round((a(n+1)-a(n))/100); all numbers 1..333 occur as these differences for a(n) < 1000000000. - Hartmut F. W. Hoft, May 18 2017

			30097 + 6 = 30103.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A158277, A046117, A023201.

Transpose[Select[Partition[Prime[Range[5000]],2,1],#[[2]]-#[[1]] == 6 && Floor[#[[1]]/100]!=Floor[#[[2]]/100]&]][[1]] (* Harvey P. Dale, Apr 28 2012 *)
a160370[n_] := Select[Range[97, n, 100], AllTrue[# + {0, 6}, PrimeQ] && NoneTrue[# + {2, 4}, PrimeQ]&]
a160370[49000] (* data *) (* Hartmut F. W. Hoft, May 18 2017 *)

{A031924(n): [A031924(n)/100] <> [A031925(n)/100]} where [..]=floor(..).

Edited by R. J. Mathar, May 14 2009

A092146 Primes of the form p + 10 where p is a prime.

Original entry on oeis.org

13, 17, 23, 29, 41, 47, 53, 71, 83, 89, 107, 113, 137, 149, 167, 173, 191, 233, 239, 251, 281, 293, 317, 347, 359, 383, 389, 419, 431, 443, 449, 467, 509, 557, 587, 617, 641, 653, 683, 701, 719, 743, 761, 797, 821, 839, 863, 887, 929, 947, 977, 1019, 1031

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Mar 31 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A006512, A023203, A046132, A046117, A092402.

lst={};Do[p=Prime[n];If[PrimeQ[p=p+10],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Prime[Range[200]]+10,PrimeQ] (* Harvey P. Dale, Aug 03 2018 *)

a(n) = 10 + A023203(n). - Alois P. Heinz, Feb 27 2020

cp's OEIS Frontend

A023201 Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)

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A023271 Primes p such that p, p+6, p+12, p+18 are all primes.

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A087695 Numbers n such that n + 3 and n - 3 are both prime.

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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).

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A098412 Greatest members p of prime triples (p-6, p-4, p).

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A111192 Product of the n-th sexy prime pair.

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A098413 Greatest members p of prime triples (p-6, p-2, p).

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