A001692 -id:A001692 - cp's OEIS frontend

A065490 Exponents in expansion of constant A065463 as Product_{n>1} zeta(n)^(-a(n)).

0, 1, -1, 1, -2, 3, -4, 5, -8, 13, -18, 25, -40, 62, -90, 135, -210, 324, -492, 750, -1164, 1809, -2786, 4305, -6710, 10460, -16264, 25350, -39650, 62057, -97108, 152145, -238818, 375165, -589520, 927200, -1459960, 2300346, -3626200

Offset: 1

N. J. A. Sloane, Nov 19 2001

sign

The sequence 1,1,1,1,2,3,4,5,8,13,18,25,40,62,90,135,... appears in Lehrer-Segal on p. 285, in the following context: Let V=Sum_{k>=1} V_k be the graded vector space H_*(PC^oo)[1], which has Poincaré series [or Poincare series] p(t)=t/(1-t^2). This sequence gives the dimensions of the free graded Lie algebra L on V.

Inverse Euler transform of F(1-n) where F() is Fibonacci numbers A000045. - Michael Somos, Jul 21 2003

Alois P. Heinz, Table of n, a(n) for n = 1..2000
G. I. Lehrer, Some sequences arising at the interface of representation theory and homotopy theory
G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
N. J. A. Sloane, Transforms

Cf. A065463.

a[n_] := DivisorSum[n, (-1)^#*MoebiusMu[n/#]*(Fibonacci[#+1] + Fibonacci[# -1]-1)&]/n; Array[a, 40] (* Jean-François Alcover, Dec 03 2015, adapted from PARI *)

a(n)=if(n<1,0,sumdiv(n,d,(-1)^d*moebius(n/d)*(fibonacci(d+1)+fibonacci(d-1)-1))/n)

a(n) = (1/n)*Sum_{d|n} (-1)^d*mu(n/d)*(Fibonacci(d-1)+Fibonacci(d+1)-1). - Vladeta Jovovic, May 03 2003

a(n) ~ (-1)^n * phi^n / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 09 2019

More terms and formula from Christian G. Bower, Aug 23 2002

A065491 Exponents in expansion of constant A065479 as a product zeta(n)^(-a(n)).

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 14, 25, 48, 88, 168, 310, 590, 1103, 2092, 3945, 7500, 14216, 27102, 51627, 98694, 188766, 361936, 694565, 1335466, 2570375, 4954744, 9561045, 18473140, 35728300, 69176558, 134063535, 260062168, 504911460, 981117286, 1907939760, 3713106350

Offset: 0

N. J. A. Sloane, Nov 19 2001

nonn

Inverse Euler transform of A001045. - R. J. Mathar, Jul 26 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A065479.

a(n) ~ 2^(n+1)/n. - Vaclav Kotesovec, Oct 09 2019

More terms from R. J. Mathar, Jul 26 2010

A065467 Decimal expansion of Product_{p prime} (1 - 1/(p^4*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.97582415304766824167901143659479983...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A078084.

$MaxExtraPrecision = 500; digits = 98; terms = 500; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0, 0, 0}, LinearRecurrence[{-2, -1, 0, 0, 1, 1}, {-5, 6, -7, 8, -9, 5}, 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 (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p^4*(p+1))) \\ Amiram Eldar, Mar 13 2021

A065468 Decimal expansion of Product_{p prime} (1 - 1/(p^5*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.9885043977412469087511066238511866644...

R. J. Mathar, Hardy-Littlewood constants embedded into..., arXiv:0903.2514 [math.NT], 2009-2011, Table 5, constant Q_1^(5).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A078083.

$MaxExtraPrecision = 500; digits = 98; terms = 500; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0, 0, 0, 0}, LinearRecurrence[{-2, -1, 0, 0, 0, 1, 1}, {-6, 7, -8, 9, -10, 11, -18}, 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 (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p^5*(p+1))) \\ Amiram Eldar, Mar 13 2021

A065477 Decimal expansion of constant 5 * Pi^2 * A065476 / 48.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.743971193350374744686559607585650...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 106.

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A065476, A078088.

$MaxExtraPrecision = 500; digits = 98; terms = 500; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{0, 1, 2}, {-2, -6, -2}, terms + 10]]; r[n_Integer] := LR[[n]];  (5 Pi^2/48)*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 18 2016 *)

(5 * Pi^2 / 48) * prodeulerrat(1 - (p+2)/p^3, 1, 3) \\ Amiram Eldar, Mar 16 2021

A065478 Decimal expansion of Product_{p prime} (1 - p/(p^3-1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.57595996889294543964316337549249669...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 106.
Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 16.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 155.
Yilan Hu and Carl Pomerance, The average order of elements in the multiplicative group of a finite field, involve, Vol. 5 (2012), No. 2, 229-236. See p. 8.
Sungjin Kim, Average results on the order of a modulo p, Journal of Number Theory, Vol. 169 (2016), pp. 353-368; arXiv preprint, arXiv:1509.01752 [math.NT], 2015.
Pär Kurlberg and Carl Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra and Number Theory, Vol. 7, No. 4 (2013), pp. 981-999; alternative link.
Pieter Moree and Peter Stevenhagen, A two-variable Artin conjecture, Journal of Number Theory, Vol. 85, No. 2 (2000), pp. 291-304; arXiv preprint, arXiv:math/9912250 [math.NT], 1999.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
P. J. Stephens, An average result for Artin's conjecture, Mathematika, Vol. 16, No. 2 (1969), pp. 178-188.
P. J. Stephens, Prime divisors of second-order linear recurrences. I,, Journal of Number Theory, Vol. 8, No. 3 (1976), pp. 313-332.
Eric Weisstein's World of Mathematics, Stephens' Constant.
Wikipedia, Stephens' constant.

Cf. A078079.

$MaxExtraPrecision = 100; m0 = 200; dm = 200; digits = 101; Clear[f]; f[m_] := f[m] = (slog = Normal[Series[Log[1 - p/(p^3 - 1)], {p, Infinity, m}]]; Exp[slog] /. Power[p, n_] -> PrimeZetaP[-n] // N[#, digits+10]&); f[m = m0]; Print[m, " ", f[m]]; f[m = m + dm]; While[Print[m, " ", f[m]]; RealDigits[f[m], 10, digits+5] != RealDigits[f[m - dm], 10, digits+5], m = m + dm];RealDigits[f[m], 10, digits] // First (* Jean-François Alcover, Sep 15 2015 *)

prodeulerrat(1 - p/(p^3-1)) \\ Amiram Eldar, Mar 17 2021

A114947 Number of monic irreducible polynomials over GF(5) of degree <= n.

Original entry on oeis.org

5, 15, 55, 205, 829, 3409, 14569, 63319, 280319, 1256567, 5695487, 26039187, 119939427, 555899247, 2590404239, 12127122989, 57005914349, 268933430849, 1272801132329, 6041172226049, 28747703565329, 137119782755669, 655421041672109, 3138947897124609

Offset: 1

Gary L Mullen (mullen(AT)math.psu.edu) and Ken Hicks, Jan 06 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums of A001692. 5th column of A143328. - Alois P. Heinz, Sep 23 2008

with(numtheory):
b:= n-> add(mobius(d) *5^(n/d)/n, d=divisors(n)):
a:= n-> add(b(k), k=1..n):
seq(a(n), n=1..30); # Alois P. Heinz, Sep 23 2008

f[n_] := DivisorSum[n, MoebiusMu[#] * 5^(n/#) &] / n; Accumulate[Array[f, 30]] (* Amiram Eldar, Aug 24 2023 *)

a(n)=sum(m=1, n, 1/m* sumdiv(m, d, moebius(d)*5^(m/d) ) ); /* Joerg Arndt, Jul 04 2011 */

a(n) ~ 5^(n+1) / (4*n). - Vaclav Kotesovec, Sep 05 2014

More terms from Alois P. Heinz, Sep 23 2008

A059863 a(n) = Product_{i=3..n} (prime(i)-4).

Original entry on oeis.org

1, 1, 1, 3, 21, 189, 2457, 36855, 700245, 17506125, 472665375, 15597957375, 577124422875, 22507852492125, 967837657161375, 47424045200907375, 2608322486049905625, 148674381704844620625, 9366486047405211099375, 627554565176149143658125, 43301264997154290912410625

Offset: 1

Labos Elemer, Feb 28 2001

nonn

See A059862 for references.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

a(n) = prod(i=3, n, prime(i)-4); \\ Michel Marcus, Aug 25 2019

More terms from Michel Marcus, Aug 25 2019

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954

G. C. Greubel, Table of n, a(n) for n = 1..350
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly,66 (1959), 375-384.

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023

A065492 Exponents in expansion of constant A065480 as a product zeta(n)^(-a(n)).

Original entry on oeis.org

0, 1, -1, 2, -4, 8, -14, 25, -48, 92, -168, 310, -590, 1117, -2092, 3945, -7500, 14264, -27102, 51627, -98694, 188934, -361936, 694565, -1335466, 2570965, -4954744, 9561045, -18473140, 35730392, -69176558, 134063535, -260062168, 504918960

Offset: 0

N. J. A. Sloane, Nov 19 2001

sign

Inverse Euler transform of A077925 shifted by two places: 1, 0, 1, -1, 3, -5,... [From R. J. Mathar, Jul 26 2010]

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A065480.

nmax = 40; s = {}; For[j = 1, j <= nmax, j++, AppendTo[s, j*(1 - (-2)^(j - 1))/3 - Sum[s[[d]]*(1 - (-2)^(j - d - 1))/3, {d, j - 1}]]]; Table[Sum[If[Divisible[j, d], MoebiusMu[j/d], 0]*s[[d]], {d, 1, j}]/j, {j, nmax}] (* Vaclav Kotesovec, Jun 13 2020 *)

a(n) ~ -(-1)^n * 2^(n+1) / n. - Vaclav Kotesovec, Jun 13 2020

More terms from R. J. Mathar, Jul 26 2010

cp's OEIS Frontend

A065490 Exponents in expansion of constant A065463 as Product_{n>1} zeta(n)^(-a(n)).

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A065491 Exponents in expansion of constant A065479 as a product zeta(n)^(-a(n)).

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A065467 Decimal expansion of Product_{p prime} (1 - 1/(p^4*(p+1))).

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A065468 Decimal expansion of Product_{p prime} (1 - 1/(p^5*(p+1))).

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A065477 Decimal expansion of constant 5 * Pi^2 * A065476 / 48.

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A065478 Decimal expansion of Product_{p prime} (1 - p/(p^3-1)).

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A114947 Number of monic irreducible polynomials over GF(5) of degree <= n.

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A059863 a(n) = Product_{i=3..n} (prime(i)-4).

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