A332493 -id:A332493 - cp's OEIS frontend

A348053 Alternative version of A332493 (Skewes number for prime n-tuplets).

1369391, 87613571, 1172531, 216646267, 251331775687, 214159878489239, 523250002674163757

Offset: 2

Hugo Pfoertner, Oct 21 2021

In contrast to A332493, in which "latest occurrence" is defined as the largest numerical value of the start of the n-tuplet, the maximum of the position of the occurrence is used here. This distinction is necessary for the first time with the term a(8), because there are 3 possible patterns of 8-tuplets. The 8-tuplet p + [0, 2, 6, 8, 12, 18, 20, 26] leads to A210439(8) = 1203255673037261. Of the two remaining candidates, p + [0, 2, 6, 12, 14, 20, 24, 26] leads to the Hardy-Littlewood prediction being exceeded at the 40634356th 8-tuplet with this pattern, the initial member of which is a(8)=523250002674163757. The other pattern p + [0, 6, 8, 14, 18, 20, 24, 26] leads to the 20316822th 8-tuplet with the beginning A332493(8) = 750247439134737983.

Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Norman Luhn, Database of the smallest prime k-tuplets, compressed files.
Hugo Pfoertner, Comparison of number of octuplets needed to achieve the H-L prediction, (2021).

Cf. A210439, A332493.

a(8) from Norman Luhn, Sep 11 2021

A022010 Initial members of prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).

Original entry on oeis.org

5639, 88799, 284729, 626609, 855719, 1146779, 6560999, 7540439, 8573429, 17843459, 19089599, 24001709, 42981929, 43534019, 69156539, 74266259, 79208399, 80427029, 84104549, 87988709, 124066079, 128469149, 144214319, 157131419, 208729049, 218033729

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 179 (modulo 210). - Matt C. Anderson, May 26 2015

			a(100) = 2526962939, a(1000) = 80752495919, a(10000) = 2010407120789, a(100000) = 42609827234069, a(1000000) = 822249634821059. See illustration for asymptotic behavior. - _Hugo Pfoertner_, Jun 15 2020

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matt C. Anderson)
Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Norman Luhn, 1 million terms (zipped archive).
Hugo Pfoertner, Illustration of n/Integral_{x=2,a(n)} 1/log(x)^7 dx approaching Hardy-Littlewood bound. (2020).

Cf. A022009 (prime septuplets of the first type), A332493.
Cf. A257124 (union of this and A022009), A343637 (septuplet following 10^n).
Cf. A357889.

[p: p in PrimesUpTo(3*10^8) | forall{p+r: r in [2, 8, 12, 14, 18, 20] | IsPrime(p+r)}]; // Vincenzo Librandi, Oct 01 2015

Select[Prime[Range[2 10^8]], Union[PrimeQ[# + {2, 8, 12, 14, 18, 20}]] == {True} &] (* Vincenzo Librandi, Oct 01 2015 *)
Select[Partition[Prime[Range[12021000]],7,1],Differences[#]=={2,6,4,2,4,2}&][[All,1]] (* or *) Select[Range[179,219*10^6,210], AllTrue[ #+{0,2,8,12,14,18,20},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 04 2019 *)

forprime(p=2, 10^30, if (isprime(p+2) && isprime(p+8) && isprime(p+12) && isprime(p+14) && isprime(p+18) && isprime(p+20), print1(p", "))) \\ Altug Alkan, Oct 01 2015. [This can be made 2x faster by inserting "p%210==179 &&" before or after "if(". - M. F. Hasler, Aug 04 2021]

use ntheory ":all"; say for sieve_prime_cluster(1,1e9, 2,8,12,14,18,20); # Dana Jacobsen, Sep 30 2015

a(n) = 210*A357889(n) + 179. - Hugo Pfoertner, Nov 18 2022

More terms from a Maple program by Matt C. Anderson, Dec 05 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
```
\\ See Alexei Kourbatov link.
```

a(7) from Hugo Pfoertner, May 09 2020

a(8) from Hugo Pfoertner, Aug 26 2021

A333586 Skewes numbers for prime n-tuples p1, p2, ..., pn, with p2 - p1 = 2.

Original entry on oeis.org

1369391, 87613571, 1172531, 21432401, 204540143441, 7572964186421

Offset: 2

Hugo Pfoertner, Mar 30 2020

a(n) is the least prime p1 starting an n-tuple of consecutive primes p1, ..., pn of minimal span pn - p1, with first gap p2 - p1 = 2, such that the difference of the occurrence count of these n-tuples and the prediction by the first Hardy-Littlewood conjecture has its first sign change. When more than one such tuple exists, the n-tuple with the lexicographically earliest sequence of gaps is chosen.
These primes are called Skewes's (or Skewes) numbers for prime k-tuples in analogy to the definition for single primes. See Tóth's article for details.
a(2) is the Skewes number for twin primes, first computed by Wolf (2011).
The minimal span s(n) = pn - p1 of the n-tuples with an initial gap of 2 is s(2) = 2, s(3) = 6, s(4) = 8, s(5) = 12, s(6) = 18, s(7) = 20, s(8) = 26.

			For n=6 two types of prime 6-tuples with first gap = 2 starting at p exist:
[p, p+2, p+6, p+8, p+12, p+18] and [p, p+2, p+8, p+12, p+14, p+18]. The first one has the lexicographically earlier sequence of gaps and is therefore chosen. The Hardy-Littlewood prediction for the number of such 6-tuples with p <= P is (C_6*15^5/2^13)*Integral_{x=2..P} 1/log(x)^6 dx with C_6 given in A269846. The 15049-th 6-tuple starting with a(6)=204540143441 is the first one for which n/Integral_{x=2..a(6)} 1/log(x)^6 dx = 17.29864469487 exceeds C_6*15^5/2^13 = 17.29861231158.

Hugo Pfoertner, Illustration of growth of number of 7-tuples, (2020).
László Tóth, On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood, CMST 25(3), 2019, 143-148.
Wikipedia, Skewes's number
Wikipedia, Twin prime, First Hardy-Littlewood conjecture.
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], 14 Jul 2011.

Cf. A005597, A065418, A065419, A269843, A269846, A271742, A333587.

The sequence of Skewes numbers always choosing the prime n-tuplets with minimal span, irrespective of the first gap, is A210439, and its variant A332493.

Li(x, n)=intnum(t=2, n, 1/log(t)^x);
\\ a(4)
C4=0.307494878758327093123354486071076853*(27/2); \\ A065419
\\ Start at 5 to exclude "fake" 4-tuple 3, 5, 7, 11
p1=5; p2=7; p3=11; n=0; forprime(p=13, 10^9, if(p-p1==8&&p-p2==6, n++; d=n-C4*Li(4, p3); if(d>=0, print(p1, " ", n, ">", C4*Li(4, p)); break)); p1=p2; p2=p3; p3=p);
\\ a(5)
C5=(15^4/2^11)*0.409874885088236474478781212337955277896358; \\ A269843
p1=3; p2=5; p3=7; p4=11; n=0; forprime(p=13, 10^9, if(p-p1==12&&p-p2==10, n++; d=n-C5*Li(5, p4); if(d>=0, print(p1, " ", n, ">", C5*Li(5, p)); break)); p1=p2; p2=p3; p3=p4; p4=p);

Changed title and clarified definition by Hugo Pfoertner, May 11 2020

A346996 a(n) = A022011(10^n).

Original entry on oeis.org

11, 1071322781, 45549998561, 1388974666811, 36073412603141, 820230015839231, 16469758685735471, 308147713085128991

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+8, p+12, p+18, p+20, 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 = 178.261954396542445395360788... is the specific Hardy-Littlewood constant for this prime constellation. The predicted approximate values using this relationship would be a(6) = 1.647755*10^16 and a(7) = 3.0824636*10^17.

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

Cf. A022011, A022012, A022013, A210439, A332493, A346997, A346998.

a(7) from Norman Luhn and Hugo Pfoertner, Sep 14 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

cp's OEIS Frontend

A348053 Alternative version of A332493 (Skewes number for prime n-tuplets).

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A022010 Initial members of prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).

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A210439 The minimal Skewes number for prime n-tuplets.

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A333586 Skewes numbers for prime n-tuples p1, p2, ..., pn, with p2 - p1 = 2.

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A346996 a(n) = A022011(10^n).

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

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

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