A005597 -id:A005597 - cp's OEIS frontend

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

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

A271780 Decimal expansion of Product_{p odd prime} 1-2/(p*(p-1)), a constant related to Artin's conjecture in the context of quadratic fields.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 15 2016

			0.5351070126166387332839586518606321598586393391028...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

Cf. A005597, A217739 (8/Pi^2).

digits = 99; $MaxExtraPrecision = 600; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, 1, -2}, {0, 4, 6}, 2 m0]; r[n_Integer] := LR[[n]];
s[m_] := s[m] = NSum[-r[n] (PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 600] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], Print[m]; m = m + dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-2/(p*(p-1)), 1, 3) \\ Amiram Eldar, Mar 11 2021

Equals (8/Pi^2)*A005597.

A307410 Numerators of the product in the singular series.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 5, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 11, 9, 65, 15, 21, 5, 69, 1, 71, 35, 3, 17, 3, 11, 77, 3, 1, 39, 81, 5, 45

Offset: 1

Mats Granvik, Apr 07 2019

Differs from A305444 at n = 35, 65, 70, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
John Omielan, How do you compute the singular series?, Mathematics Stack Exchange.
Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See the next formula after equation 2.

Cf. A005596, A005597, A305444, A380839 (denominators).

f:= proc(n) numer(mul((p-2)/(p-1),p=select(type,numtheory:-factorset(n),odd))) end proc:
map(f, [$1..100]); # Robert Israel, Apr 07 2019

Table[Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1],2]] - 2)/Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1], 2]] - 1), {h, 1, 85}]
Numerator[%]
f[p_, e_] := If[p == 2, 1, (p-2)/(p-1)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 03 2025 *)

a(n) = my(f=factor(n)[,1]~); numerator(prod(k=1, #f, if (f[k]>2, (f[k]-2)/(f[k]-1), 1))); \\ Michel Marcus, Apr 07 2019

a(n) = numerator of Product_{p|n;p>2}(p-2)/(p-1) where p is a prime number.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A380839(k) = 2 * Product_{p prime} (1-1/(p^2-p)) = 2 * A005596 = 0.7479116272384045761094... . - Amiram Eldar, Mar 03 2025

A065646 Denominators of convergents to twin prime constant.

Original entry on oeis.org

1, 1, 2, 3, 50, 103, 256, 615, 1486, 2101, 39304, 80709, 200722, 2288651, 2489373, 4778024, 12045421, 52959708, 65005129, 1093041772, 3344130445, 7781302662, 34469341093, 731637465615, 1497744272323, 607318067756430

Offset: 0

Vladeta Jovovic, Nov 08 2001

			Convergents to twin prime constant: 0,1,1/2,2/3,33/50,68/103,169/256,406/615,981/1486,1387/2101,25947/39304,...

Cf. A005597 (decimal expansion), A065645 (continued fraction), A065647 (numerators of convergents to twin prime constant), A062270, A062271.

A065647 Numerators of convergents to twin prime constant.

Original entry on oeis.org

0, 1, 1, 2, 33, 68, 169, 406, 981, 1387, 25947, 53281, 132509, 1510880, 1643389, 3154269, 7951927, 34961977, 42913904, 721584441, 2207667227, 5136918895, 22755342807, 482999117842, 988753578491, 400928198406697, 802845150391885

Offset: 0

Vladeta Jovovic, Nov 08 2001

			Convergents to twin prime constant: 0, 1, 1/2, 2/3, 33/50, 68/103, 169/256, 406/615, 981/1486, 1387/2101, 25947/39304, ...

Cf. A005597 (decimal expansion), A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A062270, A062271.

A339925 Decimal expansion of 105*zeta(3)/Pi^4.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 23 2020

			1.295730957880506136704713575049842291458572334511870477351090459267...

H. Riesel and R. C. Vaughan, On sums of primes, Ark. Mat., Volume 21, Number 1-2 (1983), 45-74 (p. 47).

Cf. A002117, A001615, A005597, A082695.

RealDigits[N[105 Zeta[3]/Pi^4, 105]][[1]]

prodeulerrat(1+1/(p*(p-1)),1,3) \\ Hugo Pfoertner, Dec 23 2020

Equals Product_{p>=3} 1+1/(p*(p-1)) where p are successive odd primes.
Equals A082695*2/3.
Equals Sum_{k>=1} A001615(k)/k^4. - Amiram Eldar, Jan 25 2024

A172168 Decimal expansion of Sum 1/q, where q is any prime of the form m^2 + 1.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Jan 28 2010

The sum is trivially convergent because each term is less than the corresponding term of Sum_{j>=1} 1/(j^2) = (Pi^2)/6.

Eight significant digits of this constant are mentioned in A083844, which gives the number of primes of the form m^2 + 1 < 10^n.

			0.8145965717029728452...

G. L. Honaker Jr. and C. Caldwell, 0.81459657, Prime Curios!.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010, pp. 6-8.

Cf. A002496, A005597.

Sum_{q in {primes of form m^2 + 1}} 1/q = Sum_{j>=1} 1/A002496(j) = 1/2 + 1/5 + 1/17 + 1/37 + 1/101 + ...

Leading zero removed and offset adjusted by R. J. Mathar, Jan 30 2010
Corrected and extended by Robert Gerbicz, Mar 13 2010
Name improved by T. D. Noe, Mar 29 2010

A269844 Primes equal to the sum of a pair of consecutive integers which are both squarefree.

Original entry on oeis.org

5, 11, 13, 29, 43, 59, 61, 67, 83, 131, 139, 157, 173, 211, 227, 229, 277, 283, 317, 331, 347, 373, 389, 419, 421, 443, 461, 509, 547, 563, 571, 619, 643, 653, 659, 661, 691, 709, 733, 787, 797, 821, 853, 859, 877, 907, 941, 947, 997, 1019, 1021, 1069, 1091, 1093, 1109, 1123, 1163, 1181, 1213

Offset: 1

Bill McEachen, Mar 06 2016

The associated prime factors will always include 2 and 3.
Will every prime number be encountered as a prime factor from the sequence entries?
The sequence appears to share many of it terms with A001122.
What is the asymptotic behavior?
Conjecture: sequence has density A271780/2 = A005597*4/Pi^2 = 0.2675535... in the primes. - Charles R Greathouse IV, Jan 24 2018
The prime terms of A179017 (except 3). - Bill McEachen, Oct 21 2021

			277 = 138 + 139 = 2*3*23 + 139 is in the sequence since both terms are squarefree.
281 = 140 + 141 = 2^2*5*7 + 3*47 is not in the sequence since the former term is divisible by 2^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Bill McEachen, A269844_vs_A001122

Cf. A001122 (primes with primitive root 2), A179017.

Cf. A005597, A271780.

Select[Prime@ Range[3, 200], PrimeOmega@ # == PrimeNu@ # &[# (# + 1)] &@ Floor[#/2] &] (* Michael De Vlieger, Mar 07 2016 *)

genit(maxx)={for(i5=3,maxx,n=prime(i5);a=factor(floor(n/2.));b=factor(ceil(n/2.));clear=1;for(j5=1,omega(floor(n/2.)),if(a[j5,2]<>1,clear=0));
for(j7=1,omega(ceil(n/2.)),if(b[j7,2]<>1,clear=0));if(clear>0,print1(n,",")));}

is(n)=isprime(n) && issquarefree(n\2) && issquarefree(n\2+1) \\ Charles R Greathouse IV, Jan 24 2018

list(lim)=my(v=List(),t=1); forfactored(k=3,(lim+1)\2, if(vecmax(k[2][,2])>1, t=0, ; if(t && isprime(t=2*k[1]-1), listput(v,t)); t=1)); Vec(v) \\ Charles R Greathouse IV, Jan 24 2018

A269846 Decimal expansion of Hardy-Littlewood constant C_6 = Product_{p prime > 6} 1/(1-1/p)^6 (1-6/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.18661429735835839665692484794418833784007394494558930487172669...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.

B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.

Cf. A005597, A065418, A065419, A269843.

$MaxExtraPrecision = 1600; digits = 99; terms = 1600; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n - 1/5^n; LR = Join[{0, 0}, LinearRecurrence[{7, -6}, {-30, -210}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First

prodeulerrat(1/(1-1/p)^6*(1-6/p), 1, 7) \\ Amiram Eldar, Mar 11 2021

A271742 Decimal expansion of Hardy-Littlewood constant C_7 = Product_{p prime > 7} 1/(1-1/p)^7 (1-7/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.3694375103864986893231907498767507770553729138930318252910123...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.

B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.

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

$MaxExtraPrecision = 1100; digits = 99; terms = 1000; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n - 1/5^n - 1/7^n; LR = Join[{0, 0}, LinearRecurrence[ {8, -7}, {-42, -336}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[ NSum[ r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First

prodeulerrat(1/(1-1/p)^7*(1-7/p), 1, 11) \\ Amiram Eldar, Mar 11 2021

cp's OEIS Frontend

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

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A271780 Decimal expansion of Product_{p odd prime} 1-2/(p*(p-1)), a constant related to Artin's conjecture in the context of quadratic fields.

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A307410 Numerators of the product in the singular series.

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A065646 Denominators of convergents to twin prime constant.

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A065647 Numerators of convergents to twin prime constant.

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A339925 Decimal expansion of 105*zeta(3)/Pi^4.

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A172168 Decimal expansion of Sum 1/q, where q is any prime of the form m^2 + 1.

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A269844 Primes equal to the sum of a pair of consecutive integers which are both squarefree.

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