A007530 -id:A007530 - cp's OEIS frontend

A248166 Prime quadruples: a(n) = A007530(1000n).

11721791, 31210841, 54112601, 78984791, 106583831, 136466501, 165939791, 196512551, 230794301, 265201421, 301001081, 335176481, 373855121, 411947141, 449008031, 486394031, 527311061, 568016921, 605541611, 648107891, 688455791, 731157611, 771988571, 814329401, 855214271, 896639321, 939594371, 983009771, 1028559401, 1073336981, 1116991871, 1162958981, 1208506331

Offset: 1

Zak Seidov, Oct 16 2014

nonn

All terms are congruent to {11,101,191} mod 210.

First 10 terms coincide with values due to M. F. Hasler (see A007530).

Zak Seidov, Table of n, a(n) for n = 1..168

Cf. A007530.

A088268 Palindromes in A007530.

Original entry on oeis.org

5, 11, 101, 191, 16061, 1748471, 179868971, 11027472011, 13160106131, 15130003151, 15843234851, 16420302461, 16782228761, 16861316861, 17060706071, 17374347371, 18491419481, 18697979681, 19497279491

Offset: 1

Amarnath Murthy, Sep 28 2003

Conjecture: Sequence is finite.

Cf. A007530.

Select[Prime[Range[8611*10^5]],AllTrue[#+{2,6,8},PrimeQ] && PalindromeQ[ #]&] (* Requires Mathematica version 10 or later *) (* The program will take a long time to run *) (* Harvey P. Dale, Mar 11 2019 *)

More terms from David Wasserman, Jul 28 2005

A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2p+q)/5 is a prime and (r+2s)/5^n is a prime.

Original entry on oeis.org

11, 101, 243701, 6758951, 3257480201, 5493848951, 58634348951, 218007942701, 21840280598951, 213065296223951, 186522444661451, 383378987630201, 7794174397786451, 110420241292317701, 67327687581380201, 91455128987630201, 3987035878499348951, 80659241994222005201, 4289429982503255201

Offset: 1

J. M. Bergot and Robert Israel, Nov 02 2022

nonn

			a(3) = 243701 because p = 247301, q = p+2 = 247303, r = p+6 = 243707, s = p+8 = 243709, (2*p+q)/5 = 146221 and (r+2*s)/5^3 = 5849 are primes, and p is the least prime that works.

Cf. A007530, A358149.

f:= proc(n) local t,p,m;
        m:= 5^n;
        t:= 3;
        do
          t:= nextprime(t);
          if t*m mod 3 <> 1 then next fi;
          p:= (t*m-22)/3;
          if isprime(p) and isprime(p+2) and isprime(p+6) and isprime(p+8) and isprime((3*p+2)/5) then return p fi;
        od;
end proc;
map(f, [$1..20]);

a[n_] := a[n] = Module[{t = 3, p, m = 5^n}, While[True, t = NextPrime[t]; If[Mod[t*m, 3] != 1, Continue[]]; p = (t*m - 22)/3; If[AllTrue[{p, p+2, p+6, p+8, (3p+2)/5}, PrimeQ], Return[p]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 19}] (* Jean-François Alcover, Jan 31 2023, after Maple program *)

A022006 Initial members p of prime 5-tuples (p, p+2, p+6, p+8, p+12).

Original entry on oeis.org

5, 11, 101, 1481, 16061, 19421, 21011, 22271, 43781, 55331, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 268811, 326141, 347981, 361211, 397751, 465161, 518801, 536441, 633461, 633791, 661091, 768191, 795791, 829721, 857951, 876011, 958541

Offset: 1

Warut Roonguthai

nonn

Subsequence of A007530. - R. J. Mathar, Feb 10 2013
All terms, except for the first one, are congruent to 11 (modulo 30). - Matt C. Anderson, May 22 2015
For n > 1 and p = a(n),  (p, p+2, p+6, p+8, p+12) are consecutive primes. - Zak Seidov, Jun 07 2017
A022007 is a similar sequence. - Wolfdieter Lang, Oct 06 2017

			Admissible 5-tuple guaranteeing sequence example: for prime(3) = 5 the first residue class starting with a nonnegative number and containing none of the members of (0, 2, 6, 8, 12) is 4 (mod 5). - _Wolfdieter Lang_, Oct 06 2017

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..10000 from Matt C. Anderson)
J. K. Andersen, Prime Records.
Tony Forbes and Norman Luhn, Prime k-tuplets.
Norman Luhn, Table of n, a(n) for n = 1..1000000.

Cf. A000040, A022007.

[p: p in PrimesUpTo(2*10^6) | IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8) and IsPrime(p+12)]; // Vincenzo Librandi, May 23 2015

lst={};Do[p=Prime[n];If[PrimeQ[p+2]&&PrimeQ[p+6]&&PrimeQ[p+8]&&PrimeQ[p+12], AppendTo[lst, p]], {n, 9!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)
Transpose[Select[Partition[Prime[Range[64000]],5,1],Differences[#] == {2,4,2,4}&]][[1]] (* Harvey P. Dale, Dec 08 2014 *)

forprime(p=2,1e7, if(isprime(p+2) && isprime(p+6) && isprime(p+8) && isprime(p+12), print1(p", "))) \\ Charles R Greathouse IV, Jul 19 2011

use ntheory ":all"; say for sieve_prime_cluster(1,1e7, 2,6,8,12); # Dana Jacobsen, Sep 30 2015

from sympy import primerange
def aupto(limit):
  p, q, r, s, alst = 2, 3, 5, 7, []
  for t in primerange(11, limit+13):
    if p+2 == q and p+6 == r and p+8 == s and p+12 == t: alst.append(p)
    p, q, r, s = q, r, s, t
  return alst
print(aupto(10**6)) # Michael S. Branicky, May 11 2021

Missing terms a(51) and a(52) added in b-file by Dana Jacobsen, Sep 30 2015

A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

Original entry on oeis.org

1, 10, 19, 82, 148, 187, 208, 325, 346, 565, 943, 1300, 1564, 1573, 1606, 1804, 1891, 1942, 2101, 2227, 2530, 3172, 3484, 4378, 5134, 5533, 6298, 6721, 6949, 7222, 7726, 7969, 8104, 8272, 8881, 9784, 9913, 10111, 10984, 11653, 11929, 12220, 13546, 14416, 15727

Offset: 1

N. J. A. Sloane and J. H. Conway, Mar 15 1996

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A024912, A102338, A102342, A102700, A007530, A014561, A008471, A032352, A216292, A216293, A125855.

Cf. A010051, A245304, A245305.

a007811 n = a007811_list !! (n-1)
a007811_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [10, 10, 10, 10]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n: n in [0..10000] | forall{10*n+r: r in [1,3,7,9] | IsPrime(10*n+r)}]; // Bruno Berselli, Sep 04 2012

for n from 1 to 10000 do m := 10*n: if isprime(m+1) and isprime(m+3) and isprime(m+7) and isprime(m+9) then print(n); fi: od: quit

Select[ Range[ 1, 10000, 3 ], PrimeQ[ 10*#+1 ] && PrimeQ[ 10*#+3 ] && PrimeQ[ 10*#+7 ] && PrimeQ[ 10*#+9 ]& ]
Select[Range[15000], And @@ PrimeQ /@ ({1, 3, 7, 9} + 10#) &] (* Ray Chandler, Jan 12 2007 *)

p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==8 && r-p==6 && q-p==2 && p%10==1, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Mar 21 2013

use ntheory ":all"; my @s = map { ($-1)/10 } sieve_prime_cluster(10,1e9, 2,6,8); say for @s; # _Dana Jacobsen, May 04 2017

a(n) = 3*A014561(n) + 1. - Zak Seidov, Sep 21 2009

A023202 Primes p such that p + 8 is also prime.

Original entry on oeis.org

3, 5, 11, 23, 29, 53, 59, 71, 89, 101, 131, 149, 173, 191, 233, 263, 269, 359, 389, 401, 431, 449, 479, 491, 563, 569, 593, 599, 653, 683, 701, 719, 743, 761, 821, 911, 929, 983, 1013, 1031, 1061, 1109, 1163, 1193, 1223, 1229, 1283, 1289, 1319, 1373, 1439

Offset: 1

David W. Wilson

All terms > 3 are congruent to 5 mod 6 (observation by Zak Seidov in SeqFan). Thus each corresponding p + 8 is congruent to 1 mod 6. - Rick L. Shepherd, Mar 25 2023

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, corrected by Sean A. Irvine and Georg Fischer)
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
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, Twin Primes

Disjoint union of A007530, A031926, A049437, A049438.

Cf. A046134, A049436, A046138, A015915.

Filtered([1..1500], k-> IsPrime(k) and IsPrime(k+8)); # G. C. Greubel, Feb 07 2020

[n: n in [0..1500] | IsPrime(n) and IsPrime(n+8)]; // Vincenzo Librandi, Nov 20 2010

select(n-> isprime(n) and isprime(n+8), [`$`(1..1500)]); # G. C. Greubel, Feb 07 2020

Select[Range[1500], PrimeQ[#] && PrimeQ[#+8]&] (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+8]&] (* Harvey P. Dale, Dec 24 2020 *)

is(n)=isprime(n)&&isprime(n+8) \\ Charles R Greathouse IV, Jul 01 2013

[n for n in (1..1500) if is_prime(n) and is_prime(n+8)] # G. C. Greubel, Feb 07 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

A078847 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <= 6 (i.e., when d = 2, 4 or 6) and forming pattern = [2, 4, 6]; short notation = [246] d-pattern.

Original entry on oeis.org

17, 41, 227, 347, 641, 1091, 1277, 1427, 1487, 1607, 2687, 3527, 3917, 4001, 4127, 4637, 4787, 4931, 8231, 9461, 10331, 11777, 12107, 13901, 14627, 20747, 21557, 23741, 25577, 26681, 26711, 27737, 27941, 28277, 29021, 31247, 32057, 32297

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A022004. - R. J. Mathar, Feb 10 2013

a(n) + 12 is the greatest term in the sequence of 4 consecutive primes with 3 consecutive gaps 2, 4, 6. - Muniru A Asiru, Aug 03 2017

			17, 17+2 = 19, 17+2+4 = 23, 17+2+4+6 = 29 are consecutive primes.

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Cf. A190814[2,4,6,8], A190817[2,4,6,8,10], A190819[2,4,6,8,10,12], A190838[2,4,6,8,10,12,14]

d = Differences[Prime[Range[10000]]]; Prime[Flatten[Position[Partition[d, 3, 1], {2, 4, 6}]]] (* T. D. Noe, May 23 2011 *)
Transpose[Select[Partition[Prime[Range[10000]],4,1],Differences[#] == {2,4,6}&]][[1]] (* Harvey P. Dale, Aug 07 2013 *)

Primes p=prime(i) such that prime(i+1) = p+2, prime(i+2) = p+2+4, prime(i+3) = p+2+4+6.

Listed terms verified by Ray Chandler, Apr 20 2009

Additional cross references from Harvey P. Dale, May 10 2014

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

cp's OEIS Frontend

A248166 Prime quadruples: a(n) = A007530(1000n).

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A088268 Palindromes in A007530.

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A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2*p+q)/5 is a prime and (r+2*s)/5^n is a prime.

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A022006 Initial members p of prime 5-tuples (p, p+2, p+6, p+8, p+12).

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A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

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A023202 Primes p such that p + 8 is also prime.

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GAP

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

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A257124 Initial members of prime septuplets.

A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2p+q)/5 is a prime and (r+2s)/5^n is a prime.