A022546 -id:A022546 - cp's OEIS frontend

A022545 Initial members of prime nonuplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26, p+30).

11, 182403491, 226449521, 910935911, 1042090781, 1459270271, 2843348351, 6394117181, 6765896981, 8247812381, 8750853101, 11076719651, 12850665671, 17140322651, 22784826131, 24816950771, 33081664151

Offset: 1

Warut Roonguthai

nonn

All terms congruent to 11 (modulo 210). - Matt C. Anderson, May 27 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 401 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, The first 10^6 initial members of prime 9-tuplets | pattern: d= 0, 2, 6, 8, 12, 18, 20, 26, 30, zip archive.

Cf. A022546, A022547, A022548.

[p: p in PrimesUpTo(250000000) | forall{p+r: r in [2, 6,8,12,18,20,26,30] | IsPrime(p+r)}]; // Vincenzo Librandi, May 27 2015

Select[Prime[Range[250000000]], Union[PrimeQ[ # +{2, 6, 8, 12, 18, 20, 26, 30}]]=={True} &] (* Vincenzo Librandi, May 27 2015 *)

forprime(p=2, 10^30, if (isprime(p+2) && isprime(p+6) && isprime(p+8) && isprime(p+12) && isprime(p+18) && isprime(p+20) && isprime(p+26) && isprime(p+30), print1(p", "))) \\ Altug Alkan, Sep 30 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e11, 2,6,8,12,18,20,26,30); # Dana Jacobsen, Sep 30 2015

More terms from Matt C. Anderson

A022547 Initial members of prime nonuplets (p, p+4, p+6, p+10, p+16, p+18, p+24, p+28, p+30).

Original entry on oeis.org

13, 113143, 626927443, 2335215973, 3447123283, 4086982633, 4422726013, 6318867403, 7093284043, 8541306853, 10998082213, 14005112893, 18869466373, 21528117883, 21843411823, 28156779793, 30303283243

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 13 (modulo 30). - Matt C. Anderson, May 28 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 1000 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, The first 10^6 initial members of prime 9-tuplets | pattern: d= 0, 4, 6, 10, 16, 18, 24, 28, 30, zip archive.

Cf. A022545, A022546, A022548.

[p: p in PrimesUpTo(2*10^8) | forall{p+r: r in [4,6,10,16,18,24,28,30] | IsPrime(p+r)}]; // Vincenzo Librandi, Sep 30 2015

composite_small := proc (n::integer)
description "determine if n has a prime factor less than 100";
if igcd(2305567963945518424753102147331756070, n) = 1 then return false else return true end if;
end proc:
# A prime constellation pattern of length 9
p := [0, 4, 6, 10, 16, 18, 24, 28, 30];
# using isprime(m*n+o+p)
o := [1273, 2263, 2683, 4003, 4633, 4993, 5893, 6883, 6943, 8623, 9613, 10243, 11563, 12823, 14863, 15133, 15553, 17863, 18433, 19753, 21163, 21793, 22483, 23053, 23113, 24103, 25783, 27733, 28723, 29983]:
with(ArrayTools):
os := Size(o, 2):
m := 30030:
loopstop := 10^11:
loopstart := 0:
print(13);
for n from loopstart to loopstop do
for a from 1 to os do
counter := 0; wc := 0; wd := 0;
while `and`(wd > -10, wd < 9) do
wd := wd+1;
if composite_small(m*n+o[a]+p[wd]) = false then wd := wd+1 else wd := -10 end if
end do;
if wd >= 9 then
while `and`(counter >= 0, wc < 9) do
wc := wc+1; if isprime(m*n+o[a]+p[wc]) then counter := counter+1 else counter := -1 end if;
end do;
end if;
if counter = 9 then print(m*n+o[a]) end if;
end do:
end do:
# Matt C. Anderson, Feb 01 2014

Select[Prime[Range[200000]], Union[PrimeQ[# + {4, 6, 10, 16, 18, 24, 28, 30}]] == {True} &] (* Vincenzo Librandi, Sep 30 2015 *)

forprime(p=2, 10^30, if (isprime(p+4) && isprime(p+6) && isprime(p+10) && isprime(p+16) && isprime(p+18) && isprime(p+24) && isprime(p+28) && isprime(p+30), print1(p", "))) \\ Altug Alkan, Sep 30 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e11, 4,6,10,16,18,24,28,30); # Dana Jacobsen, Sep 30 2015

A022548 Initial members of prime nonuplets (p, p+4, p+10, p+12, p+18, p+22, p+24, p+28, p+30).

Original entry on oeis.org

88789, 855709, 74266249, 964669609, 1422475909, 2117861719, 2558211559, 2873599429, 5766036949, 6568530949, 8076004609, 9853497739, 16394542249, 21171795079, 21956291869, 22741837819, 26486447149, 27254489389

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 169 (modulo 210). - Matt C. Anderson, May 28 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 800 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, The first 10^6 initial members of prime 9-tuplets | pattern: d= 0, 4, 10, 12, 18, 22, 24, 28, 30, zip archive.

Cf. A022545, A022546, A022547.

a := 1; for b to 25 do a := a*ithprime(b) end do; a;
# so ‘a’ is the product of the primes 2 through 97
composite_small := proc (n::integer)
description "determine if n has a prime factor less than 100";
if igcd(2305567963945518424753102147331756070, n) = 1 then return false else return true end if
end proc;
# my technique involves isprime(m*n+o+p)
# with Multiplier, Number, Offset, and Pattern
p := [0, 4, 10, 12, 18, 22, 24, 28, 30];
o := [2059, 6679, 7519, 8989, 10249, 12139, 14449, 14869, 15919, 17179, 20539, 21379, 24109, 25999, 28729];
with(ArrayTools);
os := Size(o, 2);
ps := Size(p, 2);
m := 30030;
loopstop := 10^11;
loopstart := 0;
for n from loopstart to loopstop do
for a to os do
counter := 0; wc := 0; wd := 0;
while `and`(wd > -10, wd < ps) do
wd := wd+1;
if composite_small(m*n+o[a]+p[wd]) = false then wd := wd+1 else wd := -10 end if;
end do;
if wd >= 9 then
while `and`(counter >= 0, wc < ps) do
wc := wc+1;
if isprime(m*n+o[a]+p[wc]) then counter := counter+1 else counter := -1 end if;
end do;
end if;
if counter = ps then print(m*n+o[a]) end if;
end do;
end do;
# Matt C. Anderson, Feb 13 2014

Select[Prime[Range[2 10^6]], Union[PrimeQ[# + {4, 10, 12, 18, 22, 24, 28, 30}]] == {True} &] (* Vincenzo Librandi, Sep 30 2015 *)

forprime(p=2, 10^30, if (isprime(p+4) && isprime(p+10) && isprime(p+12) && isprime(p+18) && isprime(p+22) && isprime(p+24) && isprime(p+28) && isprime(p+30), print1(p", "))) \\ Altug Alkan, Sep 30 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e11, 4,10,12,18,22,24,28,30); # Dana Jacobsen, Sep 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

A213645 Initial members of prime 12-tuplets. Primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42) are all prime.

Original entry on oeis.org

11, 380284918609481, 437163765888581, 701889794782061, 980125031081081, 1277156391416021, 1487854607298791, 1833994713165731, 2115067287743141, 2325810733931801, 3056805353932061, 3252606350489381, 3360877662097841, 3501482688249431, 3595802556731501

Offset: 1

Matt C. Anderson, Jun 17 2012

nonn

All terms, except the first one, are congruent to 1271 (modulo 2310). - Matt C. Anderson, May 29 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..2807 [first 83 entries by Matt C. Anderson]
Tony Forbes and Norman Luhn, prime k-tuplets
Norman Luhn, Table of n, a(n) for n = 1..20000
Wikipedia, Prime k-tuple

Cf. A022545, A022546, A022547, and A022548 (prime 9-tuplets).

Cf. A135311, 2*A101448 (both begin with 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42).

use ntheory ":all"; say for sieve_prime_cluster(1, 10**15, 2,6,8,12,18,20,26,30,32,36,42); # Dana Jacobsen, Oct 04 2015

Corrected and extended by Dana Jacobsen, Oct 04 2015

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.

cp's OEIS Frontend

A022545 Initial members of prime nonuplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26, p+30).

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A022547 Initial members of prime nonuplets (p, p+4, p+6, p+10, p+16, p+18, p+24, p+28, p+30).

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A022548 Initial members of prime nonuplets (p, p+4, p+10, p+12, p+18, p+22, p+24, p+28, p+30).

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A257124 Initial members of prime septuplets.

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A213645 Initial members of prime 12-tuplets. Primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42) are all prime.

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

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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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