A065481 -id:A065481 - cp's OEIS frontend

A049001 a(n) = prime(n)^2 - 2.

2, 7, 23, 47, 119, 167, 287, 359, 527, 839, 959, 1367, 1679, 1847, 2207, 2807, 3479, 3719, 4487, 5039, 5327, 6239, 6887, 7919, 9407, 10199, 10607, 11447, 11879, 12767, 16127, 17159, 18767, 19319, 22199, 22799, 24647, 26567, 27887

Offset: 1

N. J. A. Sloane

Smallest numbers k such that k*prime(n)^2 + 1 is a square. - Bruno Berselli, Apr 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.

Cf. A001248, A049002, A065481.

Cf. A084920, A166010, A182200; A182174.

a049001 = subtract 2 . a001248  -- Reinhard Zumkeller, Jul 30 2015

A049001:=n->ithprime(n)^2-2; seq(A049001(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Table[Prime[n]^2-2, {n, 50}] (* Wesley Ivan Hurt, Oct 11 2013 *)

a(n) = prime(n)^2 - 2; \\ Amiram Eldar, Nov 07 2022

a(n) = A001248(n) - 2.
a(n) = A182200(n) + 1. - Wesley Ivan Hurt, Oct 11 2013
Product_{n>=1} (1 - 1/a(n)) = A065481. - Amiram Eldar, Nov 07 2022

A065178 Number of site swap patterns with 2 balls and exact period n.

Original entry on oeis.org

1, 2, 6, 15, 42, 107, 294, 780, 2128, 5781, 15918, 43885, 122010, 340323, 954394, 2685930, 7588770, 21507696, 61144062, 174283887, 498012094, 1426213191, 4092816966, 11767176070, 33890202192, 97761428205, 282424564744

Offset: 1

Antti Karttunen, Oct 19 2001

nonn

When interspersed with 0's, exponents in expansion of A065481 as a product zeta(n)^(-a(n)).

			We have one period 1 (2), two period 2 (31/13 and 40/04) and six period three 2-ball siteswaps (312, 330, 411, 420, 501, 600) (The average of the digits is always 2).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Juggling Information Service, Site Swap FAQs
M. Macauley, Braids and Juggling patterns, Thesis (Harvey Mudd College) 2003, eq. (A1).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Row 2 of A065177.

Cf. A008683, A065179, A065180, A065481.

[seq(DistSS(p,2),p=1..60)];
A065178 := proc(n)
    add( mobius(n/d)*(3^d-2^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065178(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

a[n_] := DivisorSum[n, MoebiusMu[n/#] * (3^#-2^#)&] / n; Array[a, 30] (* Jean-François Alcover, Mar 05 2016, after R. J. Mathar *)

a(n) ~ 3^n/n. - Vaclav Kotesovec, Mar 05 2016
Inverse Euler transform of A133494. - Alois P. Heinz, Jun 23 2018
G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 3*x^k))/k. - Seiichi Manyama, Apr 14 2025

A078076 Continued fraction expansion of Product_{p prime} (1 - 1/(p^2-2)).

Original entry on oeis.org

0, 2, 1, 1, 3, 54, 1, 1, 5, 2, 3, 2, 2, 2, 2, 2, 1, 3, 32, 2, 1, 2, 17, 1, 1, 1, 1, 14, 4, 1, 19, 4, 2, 3, 40, 1, 1, 2, 1, 19, 2, 3, 5324, 2, 1, 3, 2, 1, 5, 3, 1, 1, 17, 1, 1, 2, 8, 1, 2, 2, 3, 2, 2, 1, 2, 15, 1, 7, 1, 5, 2, 1, 4, 16, 272, 1, 8, 3, 1, 2, 3, 3, 39, 2, 2, 1, 4, 1, 1, 1, 2, 3

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065481 (decimal expansion).

contfrac(prodeulerrat(1 - 1/(p^2-2))) \\ Amiram Eldar, Mar 15 2021

Offset changed by Andrew Howroyd, Jul 05 2024

cp's OEIS Frontend

A049001 a(n) = prime(n)^2 - 2.

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A065178 Number of site swap patterns with 2 balls and exact period n.

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Maple

Mathematica

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A078076 Continued fraction expansion of Product_{p prime} (1 - 1/(p^2-2)).

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Author

Keywords

Crossrefs

Programs

PARI

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