A001692 -id:A001692 - cp's OEIS frontend

A006170 Number of factorization patterns of polynomials of degree n over F_5.

1, 3, 5, 11, 17, 33, 50, 89, 135, 223, 332, 531, 776, 1194, 1730, 2591, 3700, 5429, 7660, 11035, 15417, 21851, 30225, 42300, 57966, 80146, 108956, 149076, 201073, 272587, 365009, 490656, 652700, 870597, 1150875, 1524479, 2003426, 2636589, 3446140

Offset: 1

N. J. A. Sloane

nonn

R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82. (Annotated scanned copy)

Cf. A006167-A006171.

Cf. A001692.

Euler transform of sequence b(n) = sum_{d|n, A001692(d)>=n/d} 1. - Franklin T. Adams-Watters, Jun 19 2006

More terms from Franklin T. Adams-Watters, Jun 19 2006

A056290 Number of primitive (period n) n-bead necklaces with exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 24, 300, 2400, 15750, 92680, 510288, 2691600, 13793850, 69309240, 343499100, 1686135352, 8221421250, 39901776360, 193053923860, 932142850800, 4495236287850, 21664357532920, 104388118174500, 503044634004000, 2425003910574000, 11696087875731600

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

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

Cf. A001692.

Column k=5 of A254040.

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(mobius(n/d)*k^d, d=divisors(n))/n)
    end:
a:= n-> add(b(n, 5-j)*binomial(5, j)*(-1)^j, j=0..5):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 25 2015

b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^# &]/n];
a[n_] := Sum[b[n, 5 - j]*Binomial[5, j]*(-1)^j, {j, 0, 5}];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 06 2018, after Alois P. Heinz *)

sum mu(d)*A056285(n/d) where d|n.

A056301 Number of primitive (period n) n-bead necklace structures using a maximum of five different colored beads.

Original entry on oeis.org

1, 1, 2, 5, 11, 38, 122, 496, 2005, 8707, 38364, 173562, 792827, 3662800, 17034367, 79702578, 374624253, 1767881397, 8370666416, 39751064122, 189262621739, 903220020390, 4319518316898, 20697040024784

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A001692, A008683.

a(n) = Sum_{d|n} mu(d) * A056293(n/d); mu = A008683.

A059862 a(n) = Product_{i=3..n} (prime(i) - 3).

Original entry on oeis.org

1, 1, 2, 8, 64, 640, 8960, 143360, 2867200, 74547200, 2087321600, 70968934400, 2696819507200, 107872780288000, 4746402332672000, 237320116633600000, 13289926531481600000, 770815738825932800000, 49332207284859699200000, 3354590095370459545600000, 234821306675932168192000000

Offset: 1

Labos Elemer, Feb 28 2001

nonn

			For n = 6, a(6) = 640 because:
prime(1..6)-3 = (-1,0,2,4,8,10) -> (1,1,2,4,8,10)
and
1*1*2*4*8*10 = 640. [Example generalized and reformatted per observation of _Jon E. Schoenfield_ by _Harlan J. Brothers_, Jul 15 2018]

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.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
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. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

a:= proc(n) option remember;
      `if`(n<3, 1, a(n-1)*(ithprime(n)-3))
    end:
seq(a(n), n=1..21);  # Alois P. Heinz, Nov 19 2021

Join[{1, 1}, Table[Product[Prime[i] - 3, {i, 3, n}], {n, 3, 19}]] (* Harlan J. Brothers, Jul 02 2018 *)
a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 3);
Table[a[n], {n, 19}] (* Harlan J. Brothers, Jul 02 2018 *)

a(n) = prod(i=3, n, prime(i) - 3); \\ Michel Marcus, Jul 15 2018

a(1) = a(2) = 1; a(n) = a(n-1) * (prime(n) - 3) for n >= 3. - David A. Corneth, Jul 15 2018

Name clarified, offset corrected by David A. Corneth, Jul 15 2018

A065417 Exponents in expansion of rank-2 Artin constant product(1-1/(p^3-p^2), p=prime) as a product zeta(n)^(-a(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 11, 14, 20, 27, 39, 52, 75, 102, 145, 201, 286, 397, 565, 791, 1123, 1581, 2248, 3173, 4517, 6399, 9112, 12945, 18457, 26270, 37502, 53478, 76416, 109146, 156135, 223301, 319764, 457884, 656288, 940795, 1349671, 1936620

Offset: 1

N. J. A. Sloane, Nov 15 2001

nonn

Inverse Euler transform of A078012. (The inverse of 1-1/(p^3-p^2) is p^2(p-1)/(p^3-p^2-1) = 1-1/(1+p^2-p^3). Setting 1/p=x gives (1-x)/(1-x-x^3), the g.f. of A078012.) - R. J. Mathar, Jul 26 2010

			x^3 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 6*x^11 + 7*x^12 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, sequence gamma_{2,j}^(A).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Index entries for sequences related to Artin's conjecture

Cf. A065414.

read("transforms") ;
A078012 := proc(n) option remember; if n <3 then op(n+1,[1,0,0]) ; else procname(n-1)+procname(n-3) ; end if; end proc:
a078012 := [seq(A078012(n),n=1..80)] ; EULERi(%) ;
# R. J. Mathar, Jul 26 2010

A078012[n_] := A078012[n] = If[n<3, {1, 0, 0}[[n+1]], A078012[n-1] + A078012[n-3]]; a078012 = Array[A078012, m = 80];
s = {}; For[i = 1, i <= m, i++, AppendTo[s, i*a078012[[i]] - Sum[s[[d]] * a078012[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d ], 0]*s[[d]], {d, 1, i}]/i, {i, m}] (* Jean-François Alcover, Apr 15 2016, after R. J. Mathar *)

a(n) ~ r^n / n, where r = A092526 = 1.465571231876768... - Vaclav Kotesovec, Jun 13 2020

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

A065470 Decimal expansion of Product_{p prime} (1 - 1/(p*(p^2-1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.7881625000302207005769495930535...

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

Cf. A072801, A381481.

$MaxExtraPrecision = 600; digits = 98; terms = 600; P[n_] := PrimeZetaP[n];
LR = Join[{0, 0, 0}, LinearRecurrence[{0, 2, 1, -1, -1}, {-3, 0, -5, -3, -7}, 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*(p^2-1))) \\ Amiram Eldar, Mar 13 2021

A065471 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p^2-1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.90192603958708217137771520255047...

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

Cf. A078086, A138402.

$MaxExtraPrecision = 500; digits = 98; terms = 500; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0, 0}, LinearRecurrence[{0, 2, 0, 0, 0, -1}, {-4, 0, -6, 0, -12, 0}, 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^2*(p^2-1))) \\ Amiram Eldar, Mar 16 2021

A065479 Decimal expansion of Product_{p prime >= 3} (1 - 1/(p^2-p-1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.715468235989955845094774705711728...

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

Cf. A065491, A078089.

$MaxExtraPrecision = 600; digits = 98; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n;LR = LinearRecurrence[{2, 2, -3, -2}, {0, 0, -2, -3}, 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^2-p-1), 1, 3) \\ Amiram Eldar, Mar 15 2021

A065481 Decimal expansion of Product_{p prime} (1 - 1/(p^2-2)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.38894518997956192931157878976445...

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

Cf. A065178, A078076.

$MaxExtraPrecision = 2000; digits = 98; terms = 2000; P[n_] := PrimeZetaP[n ]; LR = LinearRecurrence[{0, 5, 0, -6}, {0, 0, -2, 0}, 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^2-2)) \\ Amiram Eldar, Mar 15 2021

A065486 Decimal expansion of Product_{p prime} (1 + 1/(p+1)^2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

			1.26655850147152857161454711262964...

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

Cf. A000203, A008683, A067009.

$MaxExtraPrecision = 800; digits = 99; terms = 800; P[n_] := PrimeZetaP[n]; LR = LinearRecurrence[{-3, -4, -2}, {0, 0, 2}, 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+1)^2) \\ Amiram Eldar, Mar 15 2021

Equals Sum_{k>=1} mu(k)^2/sigma(k)^2, where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022

cp's OEIS Frontend

A006170 Number of factorization patterns of polynomials of degree n over F_5.

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A056290 Number of primitive (period n) n-bead necklaces with exactly five different colored beads.

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A056301 Number of primitive (period n) n-bead necklace structures using a maximum of five different colored beads.

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A059862 a(n) = Product_{i=3..n} (prime(i) - 3).

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A065417 Exponents in expansion of rank-2 Artin constant product(1-1/(p^3-p^2), p=prime) as a product zeta(n)^(-a(n)).

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A065470 Decimal expansion of Product_{p prime} (1 - 1/(p*(p^2-1))).

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A065471 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p^2-1))).

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A065479 Decimal expansion of Product_{p prime >= 3} (1 - 1/(p^2-p-1)).

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