A022005 -id:A022005 - cp's OEIS frontend

A277775 Decimal expansion of the prime triples constant, also known as Brun's constant B_{3b} = Sum (1/p + 1/(p+4) + 1/(p+6)) as p runs through the initial members of prime triples A022005.

8, 3, 7, 1, 1, 3, 2, 1, 2, 4, 1, 1

Offset: 0

Martin Renner, Oct 29 2016

			0.837113212411...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.14, p. 133.

Thomas R. Nicely, Prime Constellations Research Project (Mar 16 2010).

Cf. A022005, A065421, A194098, A213007, A277774.

Offset corrected by Rick L. Shepherd, Nov 03 2016

A372419 Products of prime triples (p, p+4, p+6) where p = A022005(n).

Original entry on oeis.org

1001, 4199, 65231, 347261, 1009091, 1201289, 7566179, 11592209, 22027871, 29884301, 97543451, 234118799, 564234809, 627947039, 682238471, 1296207281, 2198678291, 2901757409, 3050714441, 3283574069, 4626836849, 4881262679, 5700149369, 6542683061, 6605851859, 7956061979

Offset: 1

Michael De Vlieger, Jun 29 2024

Subsequence of A120944, subsequence of A007304.

Terms are congruent to 5 (mod 6), since A022005(n) is congruent to 1 mod 6, and 5+4 and 5+6 are congruent to 5 and 1 (mod 6), respectively. The product 1*5*1 is congruent to 5 (mod 6).

			a(1) = 1001 = 7 * 11 * 13.
a(2) = 4199 = 13 * 17 * 19.
a(3) = 65231 = 37 * 41 * 43, etc.

Cf. A007304, A016969, A022005, A120944.

Map[Times @@ NextPrime[#, {0, 1, 2}] &, Select[Prime@ Range[320], AllTrue[{# + 4, # + 6}, PrimeQ] &]]

A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 37, 41, 67, 97, 101, 103, 107, 191, 193, 223, 227, 277, 307, 311, 347, 457, 461, 613, 641, 821, 823, 853, 857, 877, 881, 1087, 1091, 1277, 1297, 1301, 1423, 1427, 1447, 1481, 1483, 1487, 1607, 1663, 1693, 1783, 1867, 1871, 1873, 1993, 1997

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Or, prime(m) such that prime(m+2) = prime(m)+6. - Zak Seidov, May 07 2012

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Cf. A031924, A023201, A098414, A098415, A098424, A098416, A098417.

Union of A022004 and A022005.

[NthPrime(n): n in [1..310] | (NthPrime(n)+6) eq NthPrime(n+2)]; // Bruno Berselli, May 07 2012

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(2*i+1, i=1..floor((N+5)/2))]):locs:= select(t -> Primes[t+2]-Primes[t]=6, [$1..nops(Primes)-2]):
Primes[locs]; # Robert Israel, Apr 30 2015

ptrsQ[n_]:=PrimeQ[n+6]&&(PrimeQ[n+2]||PrimeQ[n+4])
Select[Prime[Range[400]],ptrsQ]  (* Harvey P. Dale, Mar 08 2011 *)

p=2;q=3;forprime(r=5,1e4,if(r-p==6,print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, May 07 2012

a(n) = A098415(n) - 6. - Zak Seidov, Apr 30 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.

A257129 Initial members of prime 11-tuples.

Original entry on oeis.org

11, 1418575498573, 2118274828903, 4396774576273, 6368171154193, 6953798916913, 7908189600581, 10527733922591, 12640876669691, 27899359258003, 28138953913303, 34460918582323, 38545620633251, 40362095929003, 42023308245613, 43564522846961, 44058461657443, 60268613366231, 60596839933361, 61062361183903, 71431649320301

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

It appears that this lists only starting primes for one of the A083409(11) = 2 constellations with minimal diameter A008407(11) = 36, i.e., the union of A213646 and A213647, while there are other prime 11-tuples with larger diameter. - M. F. Hasler, Dec 03 2018

Tim Johannes Ohrtmann and Dana Jacobsen, Table of n, a(n) for n = 1..13723 [This replaces an earlier b-file from Tim Johannes Ohrtmann]
Tony Forbes k-tuplets

Initial members of all of the first prime k-tuples:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruples: A007530.
prime quintuples: A086140 out of A022007, A022006.
prime sextuples: A022008.
prime septuples: A257124 out of A022009, A022010.
prime octuples: A065706 out of A022011, A022012, A022013.
prime nonuples: A257125 out of A022547, A022548, A022545, A022546.
prime 10-tuples: A257127 out of A027569, A027570.
prime 11-tuples: this sequence out of A213646, A213647.
prime 12-tuples: A257131 out of A213601, A213645.
prime 13-tuples: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuples: A257166 out of A257167, A257168.
prime 15-tuples: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuples: A257308 out of A257369, A257370.
prime 17-tuples: A257373 out of A257374, A257375, A257376, A257377.

A257131 Initial members of prime 12-tuplets.

Original entry on oeis.org

11, 1418575498567, 27899359257997, 34460918582317, 76075560855367, 186460616596327, 218021188549237, 234280497145537, 282854319391717, 345120905374087, 346117552180627, 380284918609481, 437163765888581, 604439135284057, 701889794782061, 727417501795057, 980125031081081, 1041814617748747, 1090754719898917, 1277156391416021, 1487854607298791

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

Tim Johannes Ohrtmann and Dana Jacobsen, Table of n, a(n) for n = 1..5759 [This replaces an earlier b-file from Tim Johannes Ohrtmann]
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: A257124 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: this sequence 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.

A257135 Initial members of prime 13-tuplets.

Original entry on oeis.org

11, 13, 10527733922579, 186460616596321, 1707898733581273, 3266590043460823, 4289907938811613, 4422879865247923, 5693002600430263, 7582919852522851, 7697168877290909, 7933248530182091, 10071192314217869, 10907318641689703, 11987120084474369, 15991086371740199, 20475715985020181, 21817283854511261, 21817283854511263, 22443709342850669, 28561589689237439, 31979851757518501

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

Norman Luhn, Table of n, a(n) for n = 1..30259 (terms 1..54 from Tim Johannes Ohrtmann, terms 55..5256 from Dana Jacobsen)
Tony Forbes and Norman Luhn, 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: 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: this sequence 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.

A257137 Numbers n such that n, n+4, n+6, n+10, n+16, n+18, n+24, n+28, n+30, n+34, n+40, n+46 and n+48 are all prime.

Original entry on oeis.org

13, 4289907938811613, 5693002600430263, 21817283854511263, 48290946353555023, 51165618791484133, 53094081535451893, 70219878257874463, 98633358468021313, 99142644093930373, 104814760374339133, 166784569423739203, 167841416726358493, 184601252515266523, 263331429949004353, 272039012072134243, 339094624362619243, 363319822006646623, 363760043662280383, 437335541550793003, 455289126169953193

Offset: 1

Tim Johannes Ohrtmann, Apr 16 2015

nonn

From Robert Israel, Aug 27 2015: (Start)
All terms after the first == 1483 (mod 2730).
n+4, n+16, n+28, n+46 are in A001359. (End)

Vladimir Vlesycit and Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..944 [first 21 terms from Vladimir Vlesycit, first 99 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Smallest Prime k-tuplets
Norman Luhn, Table of n, a(n) for n = 1..6305

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: 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, this sequence, 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.

is(n)=isp=isprime; isp(n) && isp(n+4) && isp(n+6) && isp(n+10) && isp(n+16) && isp(n+18) && isp(n+24) && isp(n+28) && isp(n+30) && isp(n+34) && isp(n+40) && isp(n+46) && isp(n+48) \\ Anders Hellström, Sep 05 2015

use ntheory ":all"; say for sieve_prime_cluster(1,10**16, 4,6,10,16,18,24,28,30,34,40,46,48); # Dana Jacobsen, Oct 07 2015

cp's OEIS Frontend

A277775 Decimal expansion of the prime triples constant, also known as Brun's constant B_{3b} = Sum (1/p + 1/(p+4) + 1/(p+6)) as p runs through the initial members of prime triples A022005.

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A372419 Products of prime triples (p, p+4, p+6) where p = A022005(n).

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A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

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A257124 Initial members of prime septuplets.

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

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Magma

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

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A257129 Initial members of prime 11-tuples.

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A257131 Initial members of prime 12-tuplets.

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A257135 Initial members of prime 13-tuplets.

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A257137 Numbers n such that n, n+4, n+6, n+10, n+16, n+18, n+24, n+28, n+30, n+34, n+40, n+46 and n+48 are all prime.

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