A333587 -id:A333587 - cp's OEIS frontend

A210439 The minimal Skewes number for prime n-tuplets.

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

A332493 The minimal Skewes number for prime n-tuplets, choosing the n-tuplet with latest occurrence of the first sign change relative to the Hardy-Littlewood prediction when more than one type of n-tuplets exists (A083409(n)>1) for the given n.

Original entry on oeis.org

1369391, 87613571, 1172531, 216646267, 251331775687, 214159878489239, 750247439134737983

Offset: 2

Alexei Kourbatov and Hugo Pfoertner, May 11 2020

a(n) >= A210439(n). Equals A210439(n) at n=2,4,6, i.e., at those n for which there is only one type of prime n-tuplets (admissible prime n-tuples of minimal span). The corresponding minimal span (diameter) is given by A008407(n).
See A210439 for more information, references and links.
From Hugo Pfoertner, Oct 21 2021: (Start)
There are two options for choosing a(8):
Either one interprets "latest occurrence" as the largest number of 8-tuplets before the Hardy-Littlewood (H-L) prediction is exceeded, or one selects the larger value of the first 8-tuplet term causing the first crossing.
In the first case, 40634356 8-tuplets of the type p + [0, 2, 6, 12, 14, 20, 24, 26] are required before the H-L prediction is exceeded with an 8-tuplet 523250002674163757 + [0, 2, 6, ...].
In the second case, 20316822 8-tuplets of type p + [0, 6, 8, 14, 18, 20, 24, 26] are needed to reach the first crossing of the H-L prediction. The corresponding 8-tuplet has 750247439134737983 as first term.
The interchanging is a consequence of the different H-L constants for the two tuplet types, 475.36521.. vs. 178.261954.., which have a ratio of 8/3 to one another.
Since the H-L constant for the "earliest occurrence" A210439(8) is 178.26.., this speaks in favor of a choice from the two possibilities, which uses the same H-L constant, i.e., the occurrence with the larger tuplet start and not the occurrence with the larger number of required tuplets, for which a separate sequence A348053 is created. (End)

			Denote by pi_n(x) the n-tuplet counting function, C_n the corresponding Hardy-Littlewood constant, and Li_n(x) the integral from 2 to x of (1/(log t)^n) dt.
For 7-tuples with pattern (0 2 8 12 14 18 20) we have the Skewes number p=214159878489239; this is the initial prime p in the 7-tuple where for the first time we have pi_7(p) > C_7 Li_7(p). For the other dense pattern (0 2 6 8 12 18 20), the first sign change of pi_7(x) - C_7 Li_7(x) occurs earlier, at 7572964186421. Therefore we have a(7)=214159878489239, while A210439(7)=7572964186421.

Tony Forbes and Norman Luhn, Prime k-tuplets
Alexei Kourbatov, Optimized PARI code for computing a(7)
Norman Luhn, Database of the smallest prime k-tuplets, compressed files.
Hugo Pfoertner, Illustration of growth of number of 7-tuples, (2020).
Hugo Pfoertner, Comparison of number of octuplets needed to achieve the H-L prediction, (2021).

Cf. A008407, A083409, A210439, A333586, A333587, A348053.

PARI
```
See A. Kourbatov link.
```

\\ The first result is A210439(5), the 2nd is a(5)
Li(x, n)=intnum(t=2, n, 1/log(t)^x);
G5=(15^4/2^11)*0.409874885088236474478781212337955277896358; \\ A269843
n1=0;n2=0;n1found=0;n2found=0;p1=5;p2=7;p3=11;p4=13;
forprime(p5=17,10^12,if(p5-p1==12,my(L=Li(5,p1));if(p2-p1==2,n1++;if(!n1found&&n1/L>G5,print(p1," ",p2," ",n1," ",n1/L);n1found=1),n2++;if(!n2found&&n2/L>G5,print(p1," ",p2," ",n2," ",n2/L);n2found=1)));if(n1found&&n2found,break);p1=p2;p2=p3;p3=p4;p4=p5) \\ Hugo Pfoertner, May 12 2020
\\ Code for a(7), similar to A. Kourbatov's code but much shorter.
\\ Run time approx. 2 days, prints every 1000th 7-tuple
G7=(35^6/(3*2^22))*0.36943751038649868932319074987675; \\ A271742
s=[0,2,8,12,14,18,20];
r=[809, 2069, 2909, 5639, 6689, 7529, 7739, 8999, 10259, 12149, 12359, 14459, 14879, 15929, 17189, 19289, 20549, 21389, 23909, 24119, 26009, 27479, 28529, 28739];
forstep(p0=0,10^15,30030,for(j=1,24,my(p1=p0+r[j],isp=1,L);for(k=1,7,my(p=p1+s[k]);if(!ispseudoprime(p),isp=0;break));if(isp,L=Li(7,p1);n++;if(n%1000==0||n/L>G7,print(p1," ",p1+s[#s]," ",n/L," ",n));if(n/L>G7,break(2))))) \\ Hugo Pfoertner, May 16 2020

a(8) from Norman Luhn and Hugo Pfoertner, Oct 21 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

A152052 Number of cousin primes < 10^n.

Original entry on oeis.org

2, 9, 41, 203, 1216, 8144, 58622, 440258, 3424680, 27409999, 224373161, 1870585459, 15834656003, 135779962760, 1177207270204

Offset: 1

Cino Hilliard, Nov 22 2008

The convention here is that only the lower member of a cousin prime pair be less than the selected bound 10^n.

Cousin primes, like twin primes, can be approximated by the Hardy-Littlewood formula for the number of twin primes < n. For example, the number of cousin primes < 10^12 = 1870585459 while Hardy-Littlewood gives 1870559867. The sum of cousin primes < 10^6 divided by 4 also approximates the number of cousin primes < 10^12 with 1844802199. These two methods are asymptotic to the true value as n -> infinity.

			(3,7) and (7,11) are cousin primes < 10 since 7 < 10. So 2 is the first entry in the sequence.

Cino Hilliard, Gcc Sum Cousin primes [broken link]

Variant of A080840. [R. J. Mathar, Nov 27 2008]

Cf. A152127, A333587.

A cousin prime pair is a pair of primes that differ by 4.

a(13)-a(15) from Martin Ehrenstein, Sep 03 2021

A061642 Decimal expansion of Hardy-Littlewood constant for prime quadruples.

Original entry on oeis.org

4, 1, 5, 1, 1, 8, 0, 8, 6, 3, 2, 3, 7, 4, 1, 5, 7, 5, 7, 1, 6, 5, 2, 8, 5, 5, 6, 1, 9, 5, 9, 5, 3, 7, 5, 1, 5, 7, 9, 9, 4, 1, 0, 0, 1, 9, 3, 3, 3, 9, 6, 3, 0, 3, 2, 0, 2, 7, 1, 6, 3, 3, 4, 9, 5, 2, 1, 9, 9, 8, 3, 5, 8, 5, 0, 5, 3, 5, 5, 4, 2, 9, 9, 8, 6, 8, 4, 3, 5, 7, 3, 2, 0, 3, 1, 5, 1, 6, 6, 8, 3, 3, 4, 0, 6

Offset: 1

Jason Earls, Jun 13 2001

Computed by Robert Harley.

			4.151180863237415757165285561959537515799410019333963032027163...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

Steven R. Finch, Hardy-Littlewood Constants. [Broken link]
Steven R. Finch, Hardy-Littlewood Constants. [From the Wayback machine]
Warut Roonguthai, Large Prime Quadruplets, NMBRTHRY Archives.
Eric Weisstein's World of Mathematics, Prime Quadruplet.

Cf. A065419 (constant without factor 27/2), A333586, A333587.

$MaxExtraPrecision = 1500; digits = 105; terms = 1500; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{5, -4}, {-12, -60}, terms + 10]]; r[n_Integer] := LR[[n]]; (27/2)* Exp[NSum[ r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms,WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

(27/2) * prodeulerrat((p^3)*(p-4)/((p-1)^4), 1, 5) \\ Amiram Eldar, Mar 12 2021

Equals (27/2) * Product_{p prime > 3} (p^3)*(p-4)/((p-1)^4) using 27/2 = (3*(11+13)+(17+19))/4. - Frank Ellermann, Mar 31 2020

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A210439 The minimal Skewes number for prime n-tuplets.

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A332493 The minimal Skewes number for prime n-tuplets, choosing the n-tuplet with latest occurrence of the first sign change relative to the Hardy-Littlewood prediction when more than one type of n-tuplets exists (A083409(n)>1) for the given n.

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

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A152052 Number of cousin primes < 10^n.

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A061642 Decimal expansion of Hardy-Littlewood constant for prime quadruples.

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