A007529 -id:A007529 - cp's OEIS frontend

A098416 (A007529(n) + A098415(n)) / 4.

4, 5, 7, 8, 10, 20, 22, 35, 50, 52, 53, 55, 97, 98, 113, 115, 140, 155, 157, 175, 230, 232, 308, 322, 412, 413, 428, 430, 440, 442, 545, 547, 640, 650, 652, 713, 715, 725, 742, 743, 745, 805, 833, 848, 893, 935, 937, 938, 998, 1000, 1042, 1043, 1070, 1120, 1135

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

(A007529(n),2*a(n)+A098417(n), A098415(n)) is a prime triple (p,q,r) with p

Eric Weisstein's World of Mathematics, Prime Triplet

A098417 A098414(n) - (A007529(n) + A098415(n))/2.

Original entry on oeis.org

-1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, 1, -1

Offset: 1

Reinhard Zumkeller, Sep 07 2004

sign

(A007529(n),2*A098416(n)+a(n), A098415(n)) is a prime triple (p,q,r) with p

Eric Weisstein's World of Mathematics, Prime Triplet

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

Original entry on oeis.org

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)

A022004 Initial members of prime triples (p, p+2, p+6).

Original entry on oeis.org

5, 11, 17, 41, 101, 107, 191, 227, 311, 347, 461, 641, 821, 857, 881, 1091, 1277, 1301, 1427, 1481, 1487, 1607, 1871, 1997, 2081, 2237, 2267, 2657, 2687, 3251, 3461, 3527, 3671, 3917, 4001, 4127, 4517, 4637, 4787, 4931, 4967, 5231, 5477

Offset: 1

Warut Roonguthai

Subsequence of A001359. - R. J. Mathar, Feb 10 2013
All terms are congruent to 5 (mod 6). - Matt C. Anderson, May 22 2015
Intersection of A001359 and A023201. - Zak Seidov, Mar 12 2016

Matt C. Anderson Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
T. Forbes and Norman Luhn, Prime k-tuplets
R. J. Mathar, Table of Prime Gap Constellations (2013,2024), 275 pages (no not print...)
Thomas R. Nicely, Enumeration of the prime triples (q,q+2,q+6) to 1e16.
P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 132, ex. 3.4.3. [Broken link?]
P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 132, ex. 3.4.3.
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, Prime Triplet
Index entries for primes, gaps between

Cf. A073648, A098412, A372247 (subsequence).
Cf. A001359, A023201.
Subsequence of A007529.

[ p: p in PrimesUpTo(10000) | IsPrime(p+2) and IsPrime(p+6) ] // Vincenzo Librandi, Nov 19 2010

A022004 := proc(n)
    if n= 1 then
        5;
    else
        for a from procname(n-1)+2 by 2 do
            if isprime(a) and isprime(a+2) and isprime(a+6) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 11 2012

Select[Prime[Range[1000]], PrimeQ[#+2] && PrimeQ[#+6]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&]][[1]] (* Harvey P. Dale, Dec 24 2011 *)

is(n)=isprime(n)&&isprime(n+2)&&isprime(n+6) \\ Charles R Greathouse IV, Jul 01 2013

from sympy import primerange
def aupto(limit):
  p, q, alst = 2, 3, []
  for r in primerange(5, limit+7):
    if p+2 == q and p+6 == r: alst.append(p)
    p, q = q, r
  return alst
print(aupto(5477)) # Michael S. Branicky, May 11 2021

A022005 Initial members of prime triples (p, p+4, p+6).

Original entry on oeis.org

7, 13, 37, 67, 97, 103, 193, 223, 277, 307, 457, 613, 823, 853, 877, 1087, 1297, 1423, 1447, 1483, 1663, 1693, 1783, 1867, 1873, 1993, 2083, 2137, 2377, 2683, 2707, 2797, 3163, 3253, 3457, 3463, 3847, 4153, 4513, 4783, 5227, 5413, 5437, 5647, 5653, 5737, 6547

Offset: 1

Warut Roonguthai

Subsequence of A029710. - R. J. Mathar, May 06 2017

All terms are congruent to 1 (modulo 6). - Matt C. Anderson, May 22 2015

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe).
Tony Forbes and Norman Luhn, Prime k-tuplets.
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, Prime Triplet.

Subsequence of A029710 and of A002476.
Subsequence of A007529.
Cf. A001359, A073649, A098413.

[p: p in PrimesUpTo(10000) | IsPrime(p+4) and IsPrime(p+6)]; // Vincenzo Librandi, Aug 23 2015

Select[Table[Prime[n], {n, 2000}], PrimeQ[# + 4] && PrimeQ[# + 6] &] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

select(p->isprime(p+4) && isprime(p+6), primes(1000)) \\ Charles R Greathouse IV, Mar 17 2023

A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.

Original entry on oeis.org

23, 31, 47, 53, 61, 73, 83, 131, 151, 157, 167, 173, 233, 251, 257, 263, 271, 331, 353, 367, 373, 383, 433, 443, 503, 541, 557, 563, 571, 587, 593, 601, 607, 647, 653, 677, 727, 733, 751, 941, 947, 971, 977, 991, 1013, 1033, 1063, 1097, 1103, 1117, 1123, 1181

Offset: 1

Jeff Burch

nonn

Original name: Lower prime of a difference of 6 between consecutive primes.

Conjecture: The sequence is infinite and for every n >= 7746, a(n+1) < a(n)^(1+1/n). Namely for n >= 7746, a(n)^(1/n) is a strictly decreasing function of n (See comment lines of the sequence A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014

			23 is a term as the next prime 29 = 23 + 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020
Index entries for primes, gaps between

Cf. A001359, A023201, A031925; A031924 and A007529 together give A023201.

P:=Filtered([1..1200],IsPrime);;
List(Filtered([1..Length(P)-1],i->P[i+1]-P[i]=6),k->P[k]); # Muniru A Asiru, Jan 30 2019

[p: p in PrimesUpTo(1200) | NextPrime(p)-p eq 6]; // Bruno Berselli, Apr 09 2013

A031924 := proc(n)
    option remember;
    if n = 1 then
        return 23;
    else
        p := nextprime(procname(n-1)) ;
        q := nextprime(p) ;
        while q-p <> 6 do
            p := q ;
            q := nextprime(p) ;
        end do:
        return p;
    end if;
end proc: # R. J. Mathar, Jan 23 2013

Transpose[Select[Partition[Prime[Range[200]], 2, 1], Last[#] - First[#] == 6 &]][[1]] (* Bruno Berselli, Apr 09 2013 *)

is(n)=isprime(n)&&nextprime(n+1)-n==6 \\ Charles R Greathouse IV, Mar 21 2013

apply( A031924(n,p=2,show=0,g=6)={forprime(q=p+1,, p+g!=(p=q) || (show&&print1(p-g",")) || n-- || return(p-g))}, [1..99]) \\ Use nxt(p)=A031924(1,p) to get the term following p, use show=1 to print all a(1..n), g to select a different gap. - M. F. Hasler, Jan 02 2020

New name from M. F. Hasler, Jan 02 2020

A086140 Primes p such that three (the maximum number) primes occur between p and p+12.

Original entry on oeis.org

5, 7, 11, 97, 101, 1481, 1867, 3457, 5647, 15727, 16057, 16061, 19417, 19421, 21011, 22271, 43777, 43781, 55331, 79687, 88807, 101107, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 257857, 266677, 268811, 276037, 284737, 326141, 340927

Offset: 1

Labos Elemer, Jul 29 2003

nonn

p+12 must be a prime. - Harvey P. Dale, Jun 11 2015

A086140 is the union of A022006 and A022007. By merging the two b-files I have extended the current b-file up to n=10000 (nearly n=20000 would have been possible). I add a comparison (see Links) between the frequency of prime 5-tuples and an asymptotic approximation, which is unproven but likely to be true, and based on a conjecture first published by Hardy and Littlewood in 1923. Twins, triples and quadruplets are treated as well. - Gerhard Kirchner, Dec 07 2016

			There are two types of prime 5-tuples, and both are represented in this sequence. (11, 13, 17, 19, 23) is a prime 5-tuple of the form (p, p+2, p+6, p+8, p+12), so 11 is in the sequence, and (97, 101, 103, 107, 109) is a prime 5-tuple of the form (p, p+4, p+6, p+10, p+12), so 97 is in the sequence. - _Michael B. Porter_, Dec 19 2016

Harvey P. Dale and Gerhard Kirchner, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Gerhard Kirchner, Comparison with an assumed asymptotic distribution

Cf. A031930, A046133, A086139, A086136, A022006, A022007, A001359 (twins), A007529 (triples), A007530 (quadruplets).

cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=12, k=0}; Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&&Equal[cp[s+1, s+d-1], 3], k=k+1; Print[s]], {n, 1, 100000}]
(* Second program: *)
Transpose[Select[Partition[Prime[Range[30000]],5,1],#[[5]]-#[[1]] == 12&]][[1]] (* Harvey P. Dale, Jun 11 2015 *)

A257124 Initial members of prime septuplets.

Original entry on oeis.org

11, 5639, 88799, 165701, 284729, 626609, 855719, 1068701, 1146779, 6560999, 7540439, 8573429, 11900501, 15760091, 17843459, 18504371, 19089599, 21036131, 24001709, 25658441, 39431921, 42981929, 43534019, 45002591, 67816361, 69156539, 74266259, 79208399, 80427029, 84104549, 86818211, 87988709, 93625991, 124066079

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..1990
Tony Forbes, k-tuplets

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: this sequence out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, A257377.
Cf. A343637 (distance from 10^n to the next septuplet).
Cf. A100418.

Disjoint union of A022009 and A022010. - M. F. Hasler, Aug 04 2021

A257125 Initial members of prime 9-tuplets (or nonuplets).

Original entry on oeis.org

7, 11, 13, 17, 1277, 88789, 113143, 113147, 855709, 74266249, 182403491, 226449521, 252277007, 408936947, 521481197, 626927443, 910935911, 964669609, 1042090781, 1116452627, 1209950867, 1422475909, 1459270271, 1645175087, 2117861719, 2335215973, 2558211559, 2843348351, 2873599429, 2966003057, 3447123283, 3947480417

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

Primes prime(m) such that prime(m+8) = prime(m) + 30. - Zak Seidov, Jul 06 2015

Zak Seidov, Table of n, a(n) for n = 1..651 (Essentially original b-file by Tim Johannes Ohrtmann, just added a(1)=7 and corrected EndOfFile)
Tony Forbes and Norman Luhn, Smallest Prime k-tuplets

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: this sequence out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, A257377.

[NthPrime(n): n in [0..2*10^4] | NthPrime(n+8) eq (NthPrime(n) + 30)]; // Vincenzo Librandi, Jul 08 2015

{p, q, r, s, t, u, v, w, x} = Prime@ Range@ 9; lst = {}; While[p < 1000000001, If[p + 30 == x, AppendTo[lst, p]; Print@ p]; {p, q, r, s, t, u, v, w, x} = {q, r, s, t, u, v, w, x, NextPrime@ x}]; lst (* Robert G. Wilson v, Jul 06 2015 *)
Select[Partition[Prime[Range[5 10^6]],9,1],#[[1]]+30==#[[9]]&][[;;,1]] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Jul 01 2024 *)

main(size)=v=vector(size); i=0; m=1; while(iAnders Hellström, Jul 08 2015

A257127 Initial members of prime 10-tuplets (or decaplets).

Original entry on oeis.org

11, 9853497737, 21956291867, 22741837817, 33081664151, 83122625471, 164444511587, 179590045487, 217999764107, 231255798857, 242360943257, 294920291201, 573459229151, 663903555851, 666413245007, 688697679401, 696391309697, 730121110331, 867132039857, 974275568237, 976136848847, 1002263588297

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 101 terms from Tim Johannes Ohrtmann), Mar 01 2022
Tony Forbes and Norman Luhn Prime k-tuplets
Norman Luhn, The big database of "The smallest prime k-tuplets", section "10-tuplets": up to 10^20 as of March 2022.

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime quintuplets: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: this sequence out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, A257377.

Showing 1-10 of 45 results. Next

cp's OEIS Frontend

A098416 (A007529(n) + A098415(n)) / 4.

Views

Author

Keywords

Comments

Links

A098417 A098414(n) - (A007529(n) + A098415(n))/2.

Views

Author

Keywords

Comments

Links

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

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A022004 Initial members of prime triples (p, p+2, p+6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

A022005 Initial members of prime triples (p, p+4, p+6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Extensions

A086140 Primes p such that three (the maximum number) primes occur between p and p+12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A257124 Initial members of prime septuplets.

Views

Author

Keywords