A006489 -id:A006489 - 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)

A053070 Primes p such that p-6, p and p+6 are consecutive primes.

Original entry on oeis.org

53, 157, 173, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4457, 4597, 4657, 4993, 5107, 5113, 5303, 5387, 5393, 5563, 5807, 6073, 6263

Offset: 1

Harvey P. Dale, Feb 25 2000

Balanced primes separated from the next lower and next higher prime neighbors by 6.
Subset of A006489. - R. J. Mathar, Apr 11 2008
Subset of A006562. - Zak Seidov, Feb 14 2013
a(n) == {3,7} mod 10. - Zak Seidov, Feb 14 2013
Minimal difference is 6: a(5) - a(4) = 263 - 257, a(20) - a(19) = 1753 - 1747, ... . - Zak Seidov, Feb 14 2013

			157 is separated from both the next lower prime, 151 and the next higher prime, 163, by 6.

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

Cf. A047948, A006489, A006562. - Zak Seidov, Feb 14 2013

for i from 1 by 1 to 800 do if ithprime(i+1) = ithprime(i) + 6 and ithprime(i+2) = ithprime(i) + 12 then print(ithprime(i+1)); fi; od; # Zerinvary Lajos, Apr 27 2007

lst={};Do[p=Prime[n];If[p-Prime[n-1]==Prime[n+1]-p==6,AppendTo[lst,p]],{n,2,7!}];lst (* Vladimir Joseph Stephan Orlovsky, May 20 2010 *)
Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={6,6}&]][[2]] (* Harvey P. Dale, Oct 11 2012 *)

a(n) = A047948(n) + 6. - R. J. Mathar, Apr 11 2008

Edited by N. J. A. Sloane at the suggestion of Zak Seidov, Apr 09 2008

A141279 Primes p such that p - 6^2, p - 6, p + 6 and p + 6^2 are also primes.

Original entry on oeis.org

47, 53, 67, 73, 103, 233, 277, 353, 373, 607, 947, 977, 1187, 1223, 1487, 1663, 2683, 2693, 2713, 2963, 3307, 3733, 4457, 5443, 6323, 6863, 7523, 9007, 11903, 11933, 12107, 12547, 12583, 15313, 15767, 18217, 19427, 20107, 20753, 21523, 22073

Offset: 1

Rick L. Shepherd, Jun 21 2008

nonn

Subsequence of A006489. A141280 and A141281 are subsequences.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A006489, A141280, A141281, A141282.

pQ[n_]:=And@@PrimeQ[{n-36,n-6,n+6,n+36}]; Select[Prime[Range[10,3000]],pQ]  (* Harvey P. Dale, Feb 02 2011 *)
Select[Prime[Range[10,3000]],AllTrue[#+{-36,-6,6,36},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 05 2018 *)

A141280 Primes p such that p-6^3, p-6^2, p-6, p, p+6, p+6^2 and p+6^3 are primes.

Original entry on oeis.org

233, 353, 977, 1663, 2693, 4457, 5443, 11933, 20107, 23333, 23893, 41263, 108923, 110813, 294347, 554633, 730783, 748603, 851387, 928643, 1005013, 1008193, 1020043, 1150873, 1194763, 1326313, 1427963, 1477103, 2161337, 2212003

Offset: 1

Rick L. Shepherd, Jun 21 2008

nonn

Subsequence of A006489 and A141279. A141281 is a subsequence.

Rick L. Shepherd, Table of n, a(n) for n = 1..1000

Cf. A006489, A141279, A141281, A141282.

p6Q[n_]:=With[{c=6^Range[3]},AllTrue[Join[n+c,n-c],PrimeQ]]; Select[ Prime[ Range[ 50,200000]],p6Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 06 2015 *)

A141281 Primes p such that p-6^4, p-6^3, p-6^2, p-6, p, p+6, p+6^2, p+6^3 and p+6^4 are primes.

Original entry on oeis.org

11459317, 18726137, 73718633, 181975727, 361471043, 374195537, 419533753, 420522673, 428739323, 429198703, 456975157, 483576523, 544795393, 653578573, 682118777, 703313623, 753422317, 764967257, 797492477, 960985037, 1059913073

Offset: 1

Rick L. Shepherd, Jun 22 2008

nonn

Subsequence of A006489, A141279 and A141280. Each term is congruent to 1 or 10 mod 11 so for no prime p can this pattern be extended also to include primes p-6^5 and p+6^5 (one of them is divisible by 11). See A070392 for residues mod 11 of powers of 6. As each term of A006489 greater than 11 is congruent to 3 or 7 mod 10, combining results gives that a(n) is congruent to 23, 43, 67, or 87 mod 110.

Rick L. Shepherd, Table of n, a(n) for n = 1..55

Cf. A006489, A141279, A141280, A141282, A070392.

Select[Prime[Range[53734400]],AllTrue[#+{1296,216,36,6,-6,-36,-216,-1296},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 01 2021 *)

A161723 Middle members p of prime triples (p-18,p,p+18).

Original entry on oeis.org

23, 29, 41, 61, 71, 79, 89, 131, 149, 181, 211, 251, 331, 349, 401, 439, 449, 461, 659, 691, 701, 709, 751, 769, 839, 929, 1031, 1051, 1069, 1231, 1301, 1471, 1549, 1601, 1619, 1741, 1759, 1889, 1931, 2011, 2081, 2161, 2221, 2269, 2399, 2441, 2459, 2521

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 17 2009

nonn

The three primes p-18, p and p+18 are not necessarily consecutive.

			23 is the middle in the triple of three primes (23-18=5, 23, 23+18=41) with arithmetic progression 18.

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

Cf. A006489, A137796.

q=6*3; lst={}; Do[p=Prime[n]; If[PrimeQ[p-q] && PrimeQ[p+q], AppendTo[lst,p]], {n, 5000}]; lst
Select[Prime[Range[7,400]],AllTrue[#+{18,-18},PrimeQ]&] (* Harvey P. Dale, Apr 21 2024 *)

{p: p in A153418 and p-18 in A153418} - R. J. Mathar, Sep 22 2009

Rephrased the definition - R. J. Mathar, Sep 22 2009

A141282 Least prime p such that p-6^n, ..., p-6, p, p+6, ... and p+6^n are primes.

Original entry on oeis.org

11, 47, 233, 11459317

Offset: 1

Rick L. Shepherd, Jun 22 2008

This pattern is impossible for n >= 5. See A141281.

			a(4) = 11459317 as this is the least prime p such that p-6^4, p-6^3, p-6^2, p-6, p, p+6, p+6^2, p+6^3 and p+6^4 are all prime. The nine primes are 11458021, 11459101, 11459281, 11459311, 11459317, 11459323, 11459353, 11459533 and 11460613.

Cf. A006489, A141279, A141280, A141281.

a(1) = A006489(1), a(2) = A141279(1), a(3) = A141280(1), a(4) = A141281(1).

A161724 Primes p such that also p-24 and p+24 are primes.

Original entry on oeis.org

29, 37, 43, 47, 83, 103, 107, 113, 127, 173, 257, 293, 307, 373, 397, 433, 443, 463, 467, 523, 547, 593, 617, 677, 733, 797, 853, 863, 883, 887, 953, 1063, 1093, 1283, 1303, 1423, 1447, 1583, 1723, 1777, 1847, 1973, 2003, 2063, 2087, 2113, 2137, 2333, 2357

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 17 2009

Apart from the first term, values are 3 or 7 mod 10. - Charles R Greathouse IV, Oct 12 2009

			29-24=5,29+24=53; ...

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

Cf. A006489, A137796, A161723.

q=6*4; lst={}; Do[p=Prime[n]; If[PrimeQ[p-q] && PrimeQ[p+q], AppendTo[lst,p]], {n, 5000}]; lst

Definition edited by Emeric Deutsch, Jun 28 2009

A283562 Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 37, 43, 47, 53, 67, 73, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 223, 233, 257, 263, 277, 283, 293, 313, 337, 347, 353, 373, 397, 433, 443, 467, 487, 523, 547, 563, 577, 593, 607, 613, 617, 643, 647, 653, 733, 743, 757, 773, 787, 797, 887, 907, 937, 947, 953, 977

Offset: 1

Altug Alkan and Thomas Ordowski, Mar 11 2017

nonn

Note that p - q must be <= 12. Also note that there can be corresponding prime pairs (q, p) more than one way, i.e., (7, 13), (13, 17), (29, 31): (13^2 - 7^2)/24 = (17^2 - 13^2)/24 = (31^2 - 29^2)/24 = 5.
There are no terms of A045468 > 11.
Union of {2}, A006489, A060212, A092110, and A125272. - Robert Israel, Mar 13 2017

			3 is a term since (11^2 - 7^2)/24 = 3 and 3, 7, 11 are prime numbers.

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

Cf. A006489, A060212, A092110, A124865, A124866, A125272, A283532.

select(r -> isprime(r) and ((isprime(3*r+2) and isprime(3*r-2))
  or (isprime(6*r+1) and isprime(6*r-1))
  or (isprime(2*r+3) and isprime(2*r-3))
or (isprime(r+6) and isprime(r-6))), [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 13 2017

ok[n_] := PrimeQ[n] && Block[{p, q, s = Reduce[p^2-q^2 == 24 n && p>3 && q>3, {p, q}, Integers]}, If[s === {}, False, Or @@ And @@@ PrimeQ[{p, q} /. List@ ToRules@s]]]; Select[Range@1000, ok] (* Giovanni Resta, Mar 11 2017 *)

isA124865(n) = if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4, d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0)
lista(nn) = forprime(p=2, nn, if(isA124865(24*p), print1(p", ")))

For n > 5, a(n) == {3,7} mod 10.

A245877 Primes p such that p - d and p + d are also primes, where d is the largest digit of p.

Original entry on oeis.org

263, 563, 613, 653, 1613, 1663, 3463, 4643, 5563, 5653, 6263, 6323, 12653, 13463, 14633, 16063, 16223, 21163, 21563, 25463, 26113, 30643, 32063, 33623, 36313, 41263, 41603, 44263, 53623, 54623, 56003, 60133, 61553, 62213, 62633, 64013, 65413, 105613, 106213

Offset: 1

Colin Barker, Aug 05 2014

Intersection of A245742 and A245743.
The largest digit of a(n) is 6, and the least significant digit of a(n) is 3.
Intersection of A006489, A011536, and complements of A011537, A011538, A011539. - Robert Israel, Aug 05 2014

			The prime 263 is in the sequence because 263 - 6 = 257 and 263 + 6 = 269 are both primes.

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

Cf. A006489, A245742, A245743, A245744, A245745, A245878.

pdpQ[n_]:=Module[{m=Max[IntegerDigits[n]]},AllTrue[n+{m,-m},PrimeQ]]; Select[ Prime[Range[11000]],pdpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 13 2017 *)

select(p->d=vecsort(digits(p),,4)[1]; isprime(p-d) && isprime(p+d), primes(20000))

import sympy
from sympy import prime
from sympy import isprime
for n in range(1,10**5):
  s=prime(n)
  lst = []
  for i in str(s):
    lst.append(int(i))
  if isprime(s+max(lst)) and isprime(s-max(lst)):
    print(s,end=', ')
# Derek Orr, Aug 13 2014

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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A053070 Primes p such that p-6, p and p+6 are consecutive primes.

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A141279 Primes p such that p - 6^2, p - 6, p + 6 and p + 6^2 are also primes.

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A141280 Primes p such that p-6^3, p-6^2, p-6, p, p+6, p+6^2 and p+6^3 are primes.

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A141281 Primes p such that p-6^4, p-6^3, p-6^2, p-6, p, p+6, p+6^2, p+6^3 and p+6^4 are primes.

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A161723 Middle members p of prime triples (p-18,p,p+18).

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A141282 Least prime p such that p-6^n, ..., p-6, p, p+6, ... and p+6^n are primes.

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A161724 Primes p such that also p-24 and p+24 are primes.

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