A022012 -id:A022012 - cp's OEIS frontend

A346997 a(n) = A022012(10^n).

17, 134764997, 9844128377, 345828727877, 9637575539147, 223528482767957, 4652382265065167, 89306626080020957

Offset: 0

Hugo Pfoertner, Aug 11 2021

The terms are the (10^n)-th initial members of the prime octuplets of the form (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26). Terms a(5) and a(6) were found using a program provided by Norman Luhn during an effort to find A210439(8) and A332493(8).

Asymptotically for n -> infinity, C_HL*Integral_{x=2..a(n)} 1/log(x)^8 dx = 10^n, where C_HL = 475.36521172411318772... is the specific Hardy-Littlewood constant for this prime constellation. The predicted approximate values using this relationship would be a(6) = 4.629899*10^15 and a(7) = 8.9223552*10^16.

Norman Luhn, Number of 8-tuplets with initial members < 10^n, (2021).

Cf. A022011, A022012, A022013, A210439, A332493, A346996, A346998.

a(7) from Norman Luhn, Sep 13 2021

A347850 Record gaps between prime octuplets of the form p + {0, 2, 6, 12, 14, 20, 24, 26} (initial members are A022012), divided by 30.

Original entry on oeis.org

42, 3729, 82250, 605241, 1249017, 1734985, 1747606, 3550360, 9578800, 10562911, 12208504, 24101070, 26510262, 38121281, 38588851, 47884158, 50246371, 56392908, 59827439, 66233760, 114058040, 120197366, 141646351, 141808504, 153247005, 168751151, 235079194, 244505074

Offset: 1

Hugo Pfoertner and Norman Luhn, Sep 16 2021

nonn

			a(1) = (A022012(2) - A022012(1))/30 = (1277 - 17)/30 = 42; 17 = A347851(1).
a(5) = (A022012(13) - A022012(12))/30 = (171958667 - 165531257)/30 = 214247, exceeding all previous differences; 1655531257 = A347851(5).

Norman Luhn, Table of n, a(n) for n = 1..79
Norman Luhn, The "Smallest prime k-tuplets" database.
Norman Luhn, Record Gaps Between Prime Octuplets.

The primes at the lower end of the record gaps are given in A347851.

Cf. A022012, A346997, A347848, A347852.

A375647 Products of prime 8-tuples (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26) where p = A022012(n).

Original entry on oeis.org

435656388001, 7667061486004435747476001, 26887071293271756518203932603297162186001, 1967190066500349361284627627321478140655499961186001, 34207121652717644163491129612663352350226660003697376196001, 131790860746164880099394335252801389818740796081899944471402001

Offset: 1

Michael De Vlieger, Aug 24 2024

nonn

Primes p in A022012 belong to either 17 or 167 (mod 210).
Therefore a(n) is either congruent to the product of residues {17, 19, 23, 29, 31, 37, 41, 43} (mod 210), or {167, 169, 173, 179, 181, 187, 191, 193} (mod 210), so a(n) is congruent to 121 (mod 210).
Gaps between prime factors have a symmetric arrangement {2, 4, 6, 2, 6, 4, 2}.

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

Cf. A022012, A128470, A375646, A375648.

Map[Times @@ NextPrime[#, Range[0, 7]] &, Import["https://oeis.org/A022012/b022012.txt", "Data"][[;; 12, -1]]]

A022011 Initial members of prime octuplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26).

Original entry on oeis.org

11, 15760091, 25658441, 93625991, 182403491, 226449521, 661972301, 910935911, 1042090781, 1071322781, 1170221861, 1394025161, 1459270271, 1712750771, 1742638811, 1935587651, 2048038451, 2397437501, 2799645461

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 11 (modulo 210). - Matt C. Anderson, May 26 2015

Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 197.
Martin Gardner, Patterns in primes are a clue to the strong law of small numbers, Mathematical Games column, Scientific American, (December, 1980), pp. 20ff.

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matt C. Anderson)
T. Forbes and Norman Luhn, Prime k-tuplets
Stephan Ramon Garcia, Jeffrey Lagarias, and Ethan Simpson Lee, The error term in the truncated Perron formula for the logarithm of an L-function, arXiv:2206.01391 [math.NT], 2022.
Norman Luhn and Hugo Pfoertner, 10 million terms of A022011, 7z compressed (47.7 MB) (2021).

A065706 is the union of A022011, A022012 and A022013.
A346996(n) = a(10^n).
Cf. A347848, A347849.

[p: p in PrimesUpTo(4*10^8) | forall{p+r: r in [2,6,8,12,18,20,26] | IsPrime(p+r)}]; // Vincenzo Librandi, Oct 01 2015

Select[Prime[Range[2 10^9]], Union[PrimeQ[# + {2, 6, 8, 12, 18, 20, 26}]] == {True} &] (* Vincenzo Librandi, Oct 01 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), print1(p", "))) \\ Altug Alkan, Oct 01 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e10, 2,6,8,12,18,20,26); # Dana Jacobsen, Sep 30 2015

Reference provided by Harvey P. Dale, May 10 2013

More terms from Matt C. Anderson, Dec 06 2013

A022013 Initial members of prime octuplets (p, p+6, p+8, p+14, p+18, p+20, p+24, p+26).

Original entry on oeis.org

88793, 284723, 855713, 1146773, 6560993, 69156533, 74266253, 218033723, 261672773, 302542763, 964669613, 1340301863, 1400533223, 1422475913, 1837160183, 1962038783, 2117861723, 2249363093, 2272018733, 2558211563

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 173 (modulo 210). - Matt C. Anderson, May 26 2015

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matt C. Anderson)
T. Forbes and Norman Luhn, Prime k-tuplets
Stephan Ramon Garcia, Jeffrey Lagarias, and Ethan Simpson Lee, The error term in the truncated Perron formula for the logarithm of an L-function, arXiv:2206.01391 [math.NT], 2022.
Norman Luhn and Hugo Pfoertner, 10 million terms of A022013, 7z compressed (47.9 MB) (2021).

A065706 is the union of A022011, A022012 and A022013.
A346998(n) = a(10^n).
Cf. A347852, A347853, A357890.

[p: p in PrimesUpTo(2*10^8) | forall{p+r: r in [6,8,14,18,20,24,26] | IsPrime(p+r)}]; // Vincenzo Librandi, Sep 30 2015

Select[Prime[Range[200000]], Union[PrimeQ[# + {6, 8, 14, 18, 20, 24, 26}]] == {True} &] (* Vincenzo Librandi, Sep 30 2015 *)
Select[Prime[Range[125*10^6]],AllTrue[#+{6,8,14,18,20,24,26},PrimeQ]&] (* Harvey P. Dale, Jul 21 2025 *)

forprime(p=2, 1e30, if (isprime(p+6) && isprime(p+8) && isprime(p+14) && isprime(p+18) && isprime(p+20) && isprime(p+24) && isprime(p+26) , print1(p", "))) \\ Altug Alkan, Sep 30 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e10, 6,8,14,18,20,24,26); # Dana Jacobsen, Sep 30 2015

a(n) = 210*A357890(n) + 173. - Hugo Pfoertner, Nov 18 2022

A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.

Original entry on oeis.org

11, 17, 1277, 88793, 113147, 284723, 855713, 1146773, 2580647, 6560993, 15760091, 20737877, 25658441, 58208387, 69156533, 73373537, 74266253, 76170527, 93625991, 100658627, 134764997, 137943347, 165531257, 171958667

Offset: 1

Frank Ellermann, Dec 05 2001

nonn

3 patterns for 8-tuplets: 11010011001011, 11011010011001 and v.v.

See A022011, A022012 and A022013 for the three different possible patterns. The sequence is conjectured to be infinite, although it is not even proved that there are infinitely many twin primes (p1, p2 = p1+2). - M. F. Hasler, May 02 2015

			a(3) = 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303 = 1277+26 are primes.

Harry J. Smith and Dana Jacobsen, Table of n, a(n) for n = 1..18123 [first 100 terms from Harry J. Smith]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, The smallest prime k-tuplets, database of compressed files.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

11 = A065688(8), 26 = A008407(8), 8 = A023193(26+1), octets in A066082 are another (not minimal) constellation of 8 primes.
Union of A022011, A022012 and A022013.
See A257124 (prime septuplets) with an overview of prime k-tuplets.

{ n=0; p1=2; p8=19; for (m=1, 10^12, p1=nextprime(p1+1); p8=nextprime(p8+1); if (p8 - p1 == 26, write("b065706.txt", n++, " ", p1); if (n==100, return)) ) } \\ Harry J. Smith, Oct 26 2009

use ntheory ":all"; my($s,$e,$i,%h)=(1,1e10,0); undef @h{sieve_prime_cluster($s,$e,2,6,8,12,18,20,26), sieve_prime_cluster($s,$e,2,6,12,14,20,24,26), sieve_prime_cluster($s,$e,6,8,14,18,20,24,26)}; say ++$i," $" for sort {$a<=>$b} keys %h; # _Dana Jacobsen, Oct 10 2015

A022546 Initial members of prime nonuplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30).

Original entry on oeis.org

17, 1277, 113147, 252277007, 408936947, 521481197, 1116452627, 1209950867, 1645175087, 2966003057, 3947480417, 6234613727, 9307040837, 9853497737, 11878692167, 13766391467, 21956291867, 22741837817, 24388061207

Offset: 1

Warut Roonguthai

nonn

Subsequence of A022012. - R. J. Mathar, Feb 10 2013

All terms congruent to 17 (modulo 30). - Matt C. Anderson, May 27 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 200 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, 12, 14, 20, 24, 26, 30, zip archive.

Cf. A022545, A022547, A022548.

[p: p in PrimesUpTo(260000000) | forall{p+r: r in [2, 6, 12,14,20,24,26,30] | IsPrime(p+r)}]; // Vincenzo Librandi, May 27 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:
p := [0, 2, 6, 12, 14, 20, 24, 26, 30]:
# using isprime(m*n+o+p)
o := 17:
m:=30:
loopstop:=10^11:
loopstart:=0:
for n from loopstart to loopstop do
counter := 0:
wc := 0;
wd := 0;
while `and`(wd > -10, wd < 9) do
wd := wd+1;
if composite_small(m*n+o+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+p[wc]) then counter := counter+1 else counter := -1 end if;
end do end if;
if counter = 9 then print(m*n+o) end if;
end do:

Select[Prime[Range[260000000]], Union[PrimeQ[ # +{2, 6, 12, 14, 20, 24, 26, 30}]]=={True} &] (* Vincenzo Librandi, May 27 2015 *)

forprime(p=2, 1e30, if (isprime(p+2) && isprime(p+6) && isprime(p+12) && isprime(p+14) && isprime(p+20) && isprime(p+24) && isprime(p+26) && isprime(p+30) , print1(p", "))) \\ Altug Alkan, Sep 30 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e11, 2,6,12,14,20,24,26,30); # 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

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.

cp's OEIS Frontend

A346997 a(n) = A022012(10^n).

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A347850 Record gaps between prime octuplets of the form p + {0, 2, 6, 12, 14, 20, 24, 26} (initial members are A022012), divided by 30.

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A375647 Products of prime 8-tuples (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26) where p = A022012(n).

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A022011 Initial members of prime octuplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26).

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A022013 Initial members of prime octuplets (p, p+6, p+8, p+14, p+18, p+20, p+24, p+26).

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A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.

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A022546 Initial members of prime nonuplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30).

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A257124 Initial members of prime septuplets.

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