A065479 -id:A065479 - cp's OEIS frontend

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

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

A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

Original entry on oeis.org

1, 5, 19, 41, 109, 155, 271, 341, 505, 811, 929, 1331, 1639, 1805, 2161, 2755, 3421, 3659, 4421, 4969, 5255, 6161, 6805, 7831, 9311, 10099, 10505, 11341, 11771, 12655, 16001, 17029, 18631, 19181, 22051, 22649, 24491, 26405, 27721, 29755, 31861, 32579, 36289

Offset: 1

Kritsada Moomuang, Jan 28 2019

nonn

Terms are divisible by 5 iff p is of the form 10*m + 3 (A030431).

			a(3) = 19 because 5^2 - 5 - 1 = 19.

Robert Israel, Table of n, a(n) for n = 1..10000

Supersequence of A091568.
Subsequence of A028387 or A165900.
Second column of A378979.
Cf. A000040, A001248, A030431, A033879, A036689, A036690, A060800, A065479, A065488, A072055, A089241.
A039914 is an essentially identical sequence.

map(p -> p^2-p-1, [seq(ithprime(i),i=1..100)]); # Robert Israel, Mar 11 2019

Table[Prime[n]^2-Prime[n]-1, {n, 1, 100}] (* Jinyuan Wang, Feb 02 2019 *)

a(n) = {p=prime(n);p^2-p-1;} \\ Jinyuan Wang, Feb 02 2019

a(n) = A036689(n) - 1.
a(n) = A036690(n) - A072055(n).
a(n) = A060800(n) - A089241(n).
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065488.
Product_{n>=2} (1 - 1/a(n)) = A065479. (End)
a(n) = A033879(A001248(n)). [Deficiency of squares of primes] - Antti Karttunen, Dec 13 2024

A078089 Continued fraction expansion of Product_{p prime >= 3} (1 - 1/(p^2-p-1)).

Original entry on oeis.org

0, 1, 2, 1, 1, 16, 1, 2, 5, 4, 1, 43, 9, 1, 3, 1, 4, 2, 2, 1, 2, 1, 78, 1, 50, 3, 1, 1, 1, 2, 2, 6, 11, 4, 1, 1, 2, 1, 2, 2, 1, 4, 2, 6, 8, 1, 2, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 7, 1, 1, 154, 1, 1, 1, 1, 3, 1, 12, 1, 1, 1, 3, 1, 1, 5, 1, 2, 1, 3, 4, 1, 3, 1, 1, 1, 4, 2, 2, 10, 8, 3, 9, 1, 1, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065479 (decimal expansion).

contfrac(prodeulerrat(1 - 1/(p^2-p-1), 1, 3)) \\ Amiram Eldar, Mar 15 2021

Offset changed by Andrew Howroyd, Jul 05 2024

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

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A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

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A078089 Continued fraction expansion of Product_{p prime >= 3} (1 - 1/(p^2-p-1)).

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