A346996 -id:A346996 - cp's OEIS frontend

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

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

A210439 The minimal Skewes number for prime n-tuplets.

Original entry on oeis.org

1369391, 337867, 1172531, 21432401, 251331775687, 7572964186421, 1203255673037261

Offset: 2

Alexei Kourbatov, Jan 20 2013

More formally: the least prime in the prime n-tuplet at which for the first time pi_n(p) > C_n*Li_n(p). Here pi_n(p) is the n-tuplet counting function; C_n is the Hardy-Littlewood constant, and Li_n(x) is the integral from 2 to x of (1/(log t)^n) dt.
If, for a given n, there is more than one type of n-tuplets, then a(n) is determined by the n-tuplet type for which the first sign change of pi_n - C_n*Li_n occurs earlier than for the other type(s).
For the special case n=1, the term a(1) is the Skewes number, i.e., the first prime p for which pi(p) > Li(p). The term a(1) is not included in the sequence because it is not precisely known.

			Initially, for twin primes we have pi_2(p) < C_2 Li_2(p). The inequality is reversed for the first time for the 10744th pair of twin primes (1369391,1369393), therefore a(2) = 1369391.
Similarly, for prime triples (p,p+4,p+6), pi_3(p) < C_3 Li_3(p) until the 652nd triple (337867,337871,337873) where the inequality is reversed for the first time. Thus a(3)=337867. (The reversal for the other type of triples (p,p+2,p+6) occurs much later, so triples (p,p+2,p+6) do not contribute a term to this sequence.)
From _Hugo Pfoertner_, Aug 26 2021, Oct 24 2021: (Start)
a(8) corresponds to the 134292-th 8-tuple of the form p + [0, 2, 6, 8, 12, 18, 20, 26], found using a program provided by _Norman Luhn_. This type of 8-tuple is the one that leads to the earliest crossing of the corresponding comparison value (see linked illustration), while the other two possible configurations (enumerated in A022012 and A022013 or in A346997 and A346998) are still far from crossing their respective applicable comparison values. The other two possible 8-tuples, which lead to the crossing that occurs later, determine the terms A332493(8) and A348053(8), dependent on the criterion applied to decide what is "later". (End)

Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Tony Forbes and Norman Luhn, Prime k-tuplets.
Alexei Kourbatov, PARI code for computing a(6)
Norman Luhn, Database of the smallest prime k-tuplets, compressed files.
Hugo Pfoertner, Illustration of 8-tuplet counts determining a(8).
László Tóth, On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood, arXiv:1910.02636 [math.NT], 2019.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture
Eric Weisstein's World of Mathematics, Prime Constellation
Marek Wolf, The Skewes number for twin primes: counting sign changes of Pi_2(x)-C_2 Li_2(x), arXiv:1107.2809 [math.NT], 2011.

Cf. A052435 (round(li(n)-pi(n)), where li is the logarithmic integral and pi(x) is the prime counting function).
Cf. A332493, A333586, A333587, A348053.
Cf A022011, A022012, A022013, A346996, A346997, A346998 (related to 8-tuplets).

PARI
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\\ See Alexei Kourbatov link.
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a(7) from Hugo Pfoertner, May 09 2020

a(8) from Hugo Pfoertner, Aug 26 2021

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

Original entry on oeis.org

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

A346998 a(n) = A022013(10^n).

Original entry on oeis.org

88713, 302542763, 46328924003, 1409639621633, 37685138975573, 824339812580723, 16514635234360163, 308319877282402613

Offset: 0

Hugo Pfoertner, Aug 12 2021

The terms are the (10^n)-th initial members of the prime octuplets of the form (p, p+6, p+8, p+14, p+18, 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).

Since this prime constellation leads to the same Hardy-Littlewood constant as for A022011, the expected asymptotic behavior is also the same as in A346996 for large n. See the comment there for more information. Accordingly, the comparison value for a(6) is 1.647755*10^16 and the prediction for a(7) is 3.0824636*10^17.

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

Cf. A022011, A022012, A022013, A210439, A332493, A346996, A346997.

a(7) from Norman Luhn and Hugo Pfoertner, Sep 13 2021

A347848 Record gaps between prime octuplets of the form p + {0, 2, 6, 8, 12, 18, 20, 26} (initial members are A022011), divided by 210.

Original entry on oeis.org

75048, 323655, 422750, 2073918, 4243118, 5993757, 7828766, 11528083, 12215588, 15097361, 15513531, 17010400, 31025397, 44928642, 57138204, 75017391, 81443987, 97313005, 109587483, 110514347, 120045110, 120244355, 140472479, 140771332, 142099045, 265356757, 332391121

Offset: 1

Hugo Pfoertner and Norman Luhn, Sep 16 2021

nonn

			a(1) = (A022011(2) - A022012(1))/210 = (15760091 - 11)/210 = 75048; 11 = A347849(1).
a(5) = (A022011(21) - A022011(20))/210 = (3734403131 - 2843348351)/210 = 424318, exceeding all previous differences; 2843348351 = A347849(5).

Norman Luhn, Table of n, a(n) for n = 1..53
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 A347849.

Cf. A022011, A346996, A347850, A347852.

cp's OEIS Frontend

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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A210439 The minimal Skewes number for prime n-tuplets.

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

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A346998 a(n) = A022013(10^n).

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A347848 Record gaps between prime octuplets of the form p + {0, 2, 6, 8, 12, 18, 20, 26} (initial members are A022011), divided by 210.

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