A022008 -id:A022008 - cp's OEIS frontend

A350826 Number of prime sextuplets with n-digit initial term (A022008).

1, 1, 0, 0, 3, 0, 13, 64, 235, 1296, 7013, 41782, 253420, 1607418, 10520883, 70785653, 488096844

Offset: 1

M. F. Hasler, Jan 17 2022

Prime sextuplets are of the form (p, p+4, p+6, p+10, p+12, p+16), where p is the initial member, listed in A022008.

For n = 1 and n = 2 (see Example), the last member of the sextuplet has one digit more than the initial member (so the count would be 0 for these two, if all terms of the sextuplet had to have the same length). As far as we know, for all n > 2, all members of the sextuplets have the same length. A sufficient condition for this is that A033874(n) > 16.

			For n = 1, p = 7 is the only 1-digit prime to be the initial term of a prime sextuplet, (7, 11, 13, 17, 19, 23), hence a(1) = 1.
For n = 2, p = 97 is the only 2-digit prime to be the initial term of a prime sextuplet, (97, 101, 103, 107, 109, 113), whence a(2) = 1.
For n = 3 and n = 4, there is no n-digit prime to be the initial term of a prime sextuplet, so a(n) = 0.
For n = 5, {16057, 19417, 43777} are the only 5-digit primes which are initial members of a prime sextuplet, therefore a(5) = 3.

Norman Luhn, PI_6(10^n)

Cf. A022008 (initial members of prime sextuplets), A033874 (10^n - precprime(10^n)).

Cf. A063501, A343636.

apply( {A350826(n,L=10^n)=n=L\10; for(c=0,oo, L<(n=next_A022008(n)) && return(c))}, [1..8])

a(n) = # { p in A022008 | 10^(n-1) < p < 10^n }.

a(10)-a(12) from David A. Corneth, Jan 17 2022

a(13)-a(17) from Hugo Pfoertner, Jan 21 2022

A199004 Position of n-th prime sextuplet A022008(n) among prime 5-tuples A022007.

Original entry on oeis.org

1, 2, 7, 8, 9, 35, 49, 55, 75, 76, 82, 84, 115, 124, 133, 140, 144, 153, 167, 172, 177, 200, 201, 202, 203, 211, 213, 232, 235, 237, 240, 243, 245, 252, 256, 262, 276, 281, 285, 297, 305, 319, 322, 325, 331, 349, 352, 353, 374, 377, 379, 380, 399, 407, 415

Offset: 1

Zak Seidov, Nov 01 2011

nonn

			a(3)=7 because A022008(3)=16057=A022007(7),
a(1000)=12091 because A022008(1000)=5054057077=A022007(12091).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A022007, A022008.

lst = {}; cnt = 0; Do[p = Prime[n]; If[PrimeQ[p+4] && PrimeQ[p+6] && PrimeQ[p+10] && PrimeQ[p+12], cnt++; If[PrimeQ[p+16], AppendTo[lst, cnt]]], {n, 10000000}]; lst (* T. D. Noe, Nov 01 2011 *)

A022007(a(n))=A022008(n).

A375418 Products of prime 6-tuples (p, p+4, p+6, p+10, p+12, p+16) where p = A022008(n).

Original entry on oeis.org

7436429, 1329900201629, 17190330954910965900632429, 53723718911110731187434029, 7046153584492008675489230429, 1688812201738097614580773379554136429, 17799106117345926490096695600218208629, 55722944657811823198723449024051143429, 505827208840254150110614056219371285429

Offset: 1

Michael De Vlieger, Aug 16 2024

nonn

Subsequence of A067885.
All terms are congruent to -1 (mod 30), since they are a product of the following residues (mod 30): {7, 11, 13, 17, 19, 23}.
All terms but the first are congruent to -1 (mod 210), since they are a product of the following residues (mod 210): {97, 101, 103, 107, 109, 113}; a(1) mod 210 = 119.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022008, A067885.

Map[Times @@ NextPrime[#, Range[0, 5]] &, Select[Prime@ Range[2^20], AllTrue[{# + 4, # + 6, # + 10, # + 12, # + 16}, PrimeQ] &]]

A079010 a(n) = nextprime(16 + A022008(n)) - (16 + A022008(n)).

Original entry on oeis.org

6, 14, 14, 8, 8, 14, 18, 14, 18, 8, 24, 8, 8, 8, 18, 44, 24, 38, 18, 30, 14, 14, 8, 14, 18, 8, 8, 8, 30, 8, 38, 18, 14, 14, 66, 36, 26, 30, 30, 8, 18, 14, 8, 50, 18, 18, 14, 8, 66, 26, 14, 44, 38, 54, 18, 18, 38, 30, 8, 30, 14, 24, 26, 8, 26, 14, 8, 8, 60, 26

Offset: 1

Labos Elemer, Jan 21 2003

nonn

a(n) is the prime difference d >= 6, following [42424] difference pattern defining A022008.

			n=2: A022008(2)=97, corresponding sextuplet is {97, 101, 103, 107, 109, 113=97+16}, nextprime(113) - 113 = 127 - 113 = 14, so a(2)=14. Constraints for present terms: (a) are incongruent to 4 modulo 6; (b) Mod(a(n), 30) = {0, 6, 8, 14, 18, 20, 24, 26}; 6 occurs only once; (c) further prohibited values like e.g. 20 etc.

Cf. A022008.

d[x_] := Prime[x+1]-Prime[x]; Do[If[Equal[d[n], 4]&&Equal[d[n+1], 2]&& Equal[d[n+2], 4]&&Equal[d[n+3], 2]&& Equal[d[n+4], 4], Print[d[n+5]]], {n, 1, 100000}]

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.

cp's OEIS Frontend

A350826 Number of prime sextuplets with n-digit initial term (A022008).

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A199004 Position of n-th prime sextuplet A022008(n) among prime 5-tuples A022007.

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A375418 Products of prime 6-tuples (p, p+4, p+6, p+10, p+12, p+16) where p = A022008(n).

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A079010 a(n) = nextprime(16 + A022008(n)) - (16 + A022008(n)).

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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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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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