A269843 -id:A269843 - cp's OEIS frontend

A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.

1, 5, 10, 40, 150, 624, 2580, 11160, 48750, 217000, 976248, 4438920, 20343700, 93900240, 435959820, 2034504992, 9536718750, 44878791360, 211927516500, 1003867701480, 4768371093720, 22706531339280

Offset: 0

N. J. A. Sloane

Exponents in expansion of Hardy-Littlewood constant C_5 = 0.409874885.. = A269843 as a product_{n>=2} zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 5 colors. - Herbert Kociemba, Nov 25 2016

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1435 (terms 0..200 from T. D. Noe)
Alin Bostan, Alexander Marynych, and Kilian Raschel, On the least common multiple of several random~integers, arXiv:1901.03002 [math.PR], 2019.
Jeremie Detrey, P. J. Spaenlehauer, and P. Zimmermann, Computing the rho constant, Preprint 2016.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Dionisel Y. Regalado and Rodel Azura, An Analytic Approximation to the Density of Twin Primes, Recoletos Multidisciplinary Research Journal (2019) Vol. 6, No. 2.
G. J. Simmons, The number of irreducible polynomials of degree n over GF(p), Amer. Math. Monthly, 77 (1970), 743-745.
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
Index entries for sequences related to Lyndon words

Cf. A001037, A054720, A002105.
5th column of A074650. - Alois P. Heinz, Aug 08 2008
Cf. A008683, A000351, A027750.

a001692 n = flip div n $ sum $
            zipWith (*) (map a008683 divs) (map a000351 $ reverse divs)
            where divs = a027750_row n
-- Reinhard Zumkeller, Oct 07 2015

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*5^(n/d)/n, {d, Divisors[n]}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Mar 11 2014 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,5],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n,sumdiv(n,d,moebius(d)*5^(n/d))/n,1) \\ Charles R Greathouse IV, Jun 15 2011

a(n) = Sum_{d|n} mu(d)*5^(n/d)/n, for n>0.

G.f.: k=5, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

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

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

A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.

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

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

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