A046117 -id:A046117 - cp's OEIS frontend

A136207 Primes p such that p-6 or p+6 is prime.

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337

Offset: 1

Carlos Alves, Dec 21 2007

nonn

Either or both of (p-6) and (p+6) is/are prime. - Harvey P. Dale, Jun 22 2019

Lei Zhou, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, 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 Primes

Cf. A023201, A046117, A140546 (complement).

isA136207 := proc(n)
    if isprime(n) then
        if isprime(n+6) or isprime(n-6) then
            true;
        else
            false;
        end if;
    else
        false ;
    end if;
end proc:
A136207 := proc(n)
    option remember;
    local a;
    if n = 1 then
        5 ;
    else
        a := nextprime(procname(n-1)) ;
        while true do
            if isA136207(a) then
                return a;
            else
                a := nextprime(a) ;
            end if;
        end do:
    end if;
end proc:
seq(A136207(n),n=1..80) ; # R. J. Mathar, Jun 10 2024

dd = 6; DistancePrimesQ1 = (PrimeQ[ # ] && PrimeQ[ # + dd]) &; DistancePrimesQ2 = (PrimeQ[ # ] && PrimeQ[ # - dd] && (# > dd)) &; DistancePrimesQQ = (DistancePrimesQ1[ # ] || DistancePrimesQ2[ # ]) &; DistancePrimes = Select[Range[ # ], DistancePrimesQQ] &; DistancePrimes[1000]
Alternative by Lei Zhou:
p = 3; Table[While[p = NextPrime[p]; ! (PrimeQ[p - 6] || PrimeQ[p + 6])]; p, {n, 1, 100}]
Select[Prime[Range[3,100]],AnyTrue[#+{6,-6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2019 *)

A104229 Primes equal to the product of two successive sexy primes plus 6.

Original entry on oeis.org

61, 97, 193, 397, 673, 1153, 1597, 1933, 4093, 7393, 12097, 37633, 64513, 70753, 96097, 122497, 126733, 136897, 190093, 211597, 256033, 313597, 329473, 348097, 430333, 541693, 781453, 891133, 988033, 1267873, 1416097, 1674433, 2102497

Offset: 1

Giovanni Teofilatto, Apr 02 2005

Primes of the form 6 + A111192(i). - R. J. Mathar, Nov 26 2008

All numbers in this sequence are of the form 12n + 1. Also, as one would expect from a random distribution of sexy prime pairs, with the exception of 61, in decimal two thirds of these numbers end in 3, and the other third end in 7. - Daniel Mondot, Apr 29 2024

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A023201, A046117, A104227, A104228, A111192.

Extended by R. J. Mathar, Nov 26 2008

A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

Original entry on oeis.org

55, 91, 187, 247, 275, 391, 605, 637, 667, 1147, 1183, 1375, 1591, 1927, 2057, 2491, 3025, 3127, 3179, 3211, 4087, 4459, 4693, 4891, 5767, 6647, 6655, 6875, 7387, 8281, 8993, 9991, 10807, 11227, 12091, 15125, 15341, 15379, 17947, 19343, 22627, 23707

Offset: 1

Reinhard Zumkeller, Jul 30 2008

nonn

Subsequence of A007774.
A111192 is a subsequence.
Subsequence of A195118. - Reinhard Zumkeller, Sep 13 2011

			a(1) = 55 = 5 * 11 = A023201(1) * A046117(1).
a(2) = 91 = 7 * 13 = A023201(2) * A046117(2).
a(3) = 187 = 11 * 17 = A023201(3) * A046117(3).
a(4) = 247 = 13 * 19 = A023201(4) * A046117(4).
a(5) = 275 = 5^2 * 11 = A023201(1)^2 * A046117(1).
a(6) = 391 = 17 * 23 = A023201(5) * A046117(5).
a(7) = 605 = 5 * 11^2 = A023201(1) * A046117(1)^2.
a(8) = 637 = 7^2 * 13 = A023201(2)^2 * A046117(2).
a(9) = 667 = 23 * 29 = A023201(6) * A046117(6).
a(10) = 1147 = 31 * 37 = A023201(7) * A046117(7).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
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]
Index entries for primes, gaps between.

Cf. A001221, A006530, A007774, A020639, A023201, A027748, A046117, A111192, A143201, A195118.

a143205 n = a143205_list !! (n-1)
a143205_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 6 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

okQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[1]]},Length[fi]==2 && Last[fi]-First[fi]==6]; Select[Range[25000],okQ]  (* Harvey P. Dale, Apr 18 2011 *)

A143201(a(n)) = 7.
A020639(a(n)) in A023201 and A006530(a(n)) in A046117.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023201(n)+2)^2-9) = 0.058842810164... . - Amiram Eldar, Oct 26 2024

A156274 List of prime pairs of the form (p, p+6).

Original entry on oeis.org

5, 11, 7, 13, 11, 17, 13, 19, 17, 23, 23, 29, 31, 37, 37, 43, 41, 47, 47, 53, 53, 59, 61, 67, 67, 73, 73, 79, 83, 89, 97, 103, 101, 107, 103, 109, 107, 113, 131, 137, 151, 157, 157, 163, 167, 173, 173, 179, 191, 197, 193, 199, 223, 229, 227, 233, 233, 239, 251, 257

Offset: 1

Vincenzo Librandi, Feb 07 2009

A023201 and A046117 interleaved. [From R. J. Mathar, Feb 19 2009]

			For p=5, p+6=11, (5,11); p=7, p+6=13, (7,13); p=11, p+6=17, (11,17)

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

Flatten[Select[{#, # + 6}&/@Prime[Range[100]], PrimeQ[Last[#]]&]] (* Vincenzo Librandi, Apr 06 2013 *)

A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

Original entry on oeis.org

7, 19, 37, 67, 79, 97, 109, 127, 229, 277, 307, 349, 379, 397, 439, 457, 487, 499, 739, 757, 769, 859, 877, 907, 937, 967, 1009, 1087, 1279, 1297, 1429, 1447, 1489, 1549, 1567, 1579, 1597, 1609, 1867, 1999, 2137, 2239, 2269, 2347, 2377, 2389, 2437, 2539, 2617, 2659, 2689, 2707, 2749, 2797, 2857

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's or 9's.

			7 is in this sequence since pair (7,11) is the first with difference 4 spanning a multiple of 10.

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288022, A288024.

a288021[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==4&]]
a288021[3000] (* data *)

A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

Original entry on oeis.org

47, 157, 167, 257, 367, 557, 587, 607, 647, 677, 727, 947, 977, 1097, 1117, 1187, 1217, 1367, 1657, 1747, 1777, 1907, 1987, 2207, 2287, 2417, 2467, 2677, 2837, 2897, 2957, 3307, 3407, 3607, 3617, 3637, 3727, 3797, 4007, 4357, 4457, 4507, 4597, 4657, 4937, 4987

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's.

Number of terms < 10^k: 0, 0, 1, 13, 81, 565, 4027, 30422, 237715, ... - Muniru A Asiru, Jan 09 2018

			47 is in the sequence since pair (47,53) is the first with difference 6 spanning a multiple of 10.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288024.

P:=Filtered([1..20000], IsPrime);
P1:=List(Filtered(Filtered(List([1..Length(P)-1],n->[P[n],P[n+1]]),i->i[2]-i[1]=6),j->j[1] mod 5=2),k->k[1]); # Muniru A Asiru, Jul 08 2017

for n from 1 to 2000 do if [ithprime(n+1)-ithprime(n), ithprime(n) mod 5] = [6,2] then print(ithprime(n)); fi; od; # Muniru A Asiru, Jan 19 2018

a288022[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==6&]]
a288022[3000] (* data *)

A288024 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 8, and p1, p2 are in different decades.

Original entry on oeis.org

89, 359, 389, 449, 479, 683, 719, 743, 929, 983, 1109, 1163, 1193, 1373, 1439, 1523, 1559, 1733, 1823, 1979, 2003, 2153, 2213, 2243, 2273, 2459, 2609, 2663, 2699, 2843, 2879, 2909, 3209, 3449, 3623, 3719, 4289, 4349, 4583, 4943, 5189, 5399, 5573, 5693, 5783, 5813

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 3's or 9's.

			89 is in the sequence since pair (89,97) is the first with difference 8 spanning a multiple of 10.

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

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288022.

a288024[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==8&]]
a288024[6000] (* data *)
Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==8&&IntegerDigits[#[[1]]][[-2]]!= IntegerDigits[ #[[2]]][[-2]]&][[;;,1]] (* Harvey P. Dale, Jan 09 2024 *)

A080840 Number of cousin primes < 10^n.

Original entry on oeis.org

1, 8, 41, 203, 1216, 8144, 58622, 440258, 3424680, 27409999, 224373161, 1870585459, 15834656003, 135779962760, 1177207270204

Offset: 1

Jason Earls, Mar 28 2003

The corresponding numbers for twin primes and sexy primes are in A007508 and A080841, the greater of twin primes, cousin primes and sexy primes are in A006512, A046132 and A046117 respectively.

In this sequence, only the upper member of each prime cousin pair is counted. See A152052 for the variant where only the lower member is counted. - James Rayman, Jan 17 2021

A. Granville and G. Martin, Prime number races, Amer. Math. Monthly vol 113, no 1 (2006) p 1.
Eric Weisstein's World of Mathematics, Cousin Primes.

Cf. A007508, A080841, A006512, A046132, A046117, A152052, A152127.

{c=0; p=5; for(n=1,9, while(p<10^n,if(isprime(p-4),c++); p=nextprime(p+1)); print1(c,","))}

a(8) and a(9) from Klaus Brockhaus, Mar 30 2003
More terms from R. J. Mathar, Aug 05 2007
a(13)-a(15) from Martin Ehrenstein, Sep 03 2021

A080841 Number of pairs (p,q) of (not necessarily consecutive) primes with q-p = 6 and q < 10^n.

Original entry on oeis.org

0, 15, 74, 411, 2447, 16386, 117207, 879908, 6849047, 54818296, 448725003, 3741217498

Offset: 1

Jason Earls, Mar 28 2003

nonn

Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the latter, whereas A093738 considers the former. - N. J. A. Sloane, Mar 07 2021

A. Granville, G. Martin, Prime number races, Amer. Math. Monthly vol 113, no 1 (2006) p 1.
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, A007508, A080840, A006512, A046132, A046117.

A104228 Primes one larger than the sum over a sexy prime pair.

Original entry on oeis.org

17, 29, 41, 53, 89, 101, 113, 173, 269, 353, 389, 461, 509, 521, 701, 773, 929, 1013, 1181, 1193, 1289, 1301, 1361, 1721, 1889, 1901, 1949, 2213, 2381, 2441, 2609, 2729, 2741, 2861, 2969, 3209, 3221, 3821, 4001, 4133, 4421, 4481, 4673, 4793, 4889, 5381, 5393, 5801, 5813, 6173, 6653, 7349, 7529

Offset: 1

Giovanni Teofilatto, Apr 02 2005

Primes of the form A023201(i)+A046117(i)+1 - R. J. Mathar, Nov 26 2008

			17=5+11+1 is prime and one larger than the sum 5+11 over the first sexy prime pair. - _R. J. Mathar_, Nov 26 2008

Cf. A023201, A046117, A104227, A104229.

Inserted 89 and extended beyond a(8). - R. J. Mathar, Nov 26 2008

cp's OEIS Frontend

A136207 Primes p such that p-6 or p+6 is prime.

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Maple

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A104229 Primes equal to the product of two successive sexy primes plus 6.

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A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

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A156274 List of prime pairs of the form (p, p+6).

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A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

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A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

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A288024 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 8, and p1, p2 are in different decades.

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Mathematica

A080840 Number of cousin primes < 10^n.

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PARI