A210439 -id:A210439 - cp's OEIS frontend

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.

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

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

A347278 First member p(m) of the m-th twin prime pair such that d(m) > 0 and d(m-1) < 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx - C, C = 2*A005597 = A114907.

Original entry on oeis.org

1369391, 1371989, 1378217, 1393937, 1418117, 1426127, 1428767, 1429367, 1430291, 1494509, 1502141, 1502717, 1506611, 1510307, 35278697, 35287001, 35447171, 35468429, 35468861, 35470271, 35595869, 45274121, 45276227, 45304157, 45306827, 45324569, 45336461, 45336917

Offset: 1

Hugo Pfoertner, Aug 26 2021

nonn

The sequence gives the positions, expressed by A001359(m), where the number of twin prime pairs m seen so far first exceeds the number predicted by the first Hardy-Littlewood conjecture after having been less than the predicted number before. A347279 gives the transitions in the opposite direction.

The total number of twin prime pairs up to that with first member x in the intervals a(k) <= x < A347279(k) is above the Hardy-Littlewood prediction. The total number of twin prime pairs up to that with first member x in the intervals A347279(k) <= x < a(k+1) is below the H-L prediction.

Hugo Pfoertner, Table of n, a(n) for n = 1..12135
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. A001359, A005597, A114907, A152051, A347279.

a(1) = A210439(2) (Skewes number for twin primes).

halicon(h) = {my(w=Set(vecsort(h)), n=#w, wmin=vecmin(w), distres(v,p)=#Set(v%p)); for(k=1,n, w[k]=w[k]-wmin); my(plim=nextprime(vecmax(w))); prodeuler(p=2, plim, (1-distres(w,p)/p)/(1-1/p)^n) * prodeulerrat((1-n/p)/(1-1/p)^n, 1, nextprime(plim+1))}; \\ k-tuple constant
Li(x, n)=intnum(t=2, n, 1/log(t)^x); \\ logarithmic integral
a347278(nterms,CHL)={my(n=1,pprev=1,np=0); forprime(p=5,, if(p%6!=1&&ispseudoprime(p+2), n++; L=Li(2,p); my(x=n/L-CHL); if(x*pprev>0, if(pprev>0,print1(p,", ");np++; if(np>nterms,return)); pprev=-pprev)))};
a347278(10,halicon([0,2])) \\ computing 30 terms takes about 5 minutes

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

Original entry on oeis.org

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

cp's OEIS Frontend

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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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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A347278 First member p(m) of the m-th twin prime pair such that d(m) > 0 and d(m-1) < 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx - C, C = 2*A005597 = A114907.

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

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