A065486 -id:A065486 - cp's OEIS frontend

A110833 a(n) = (prime(n)+1)^2.

9, 16, 36, 64, 144, 196, 324, 400, 576, 900, 1024, 1444, 1764, 1936, 2304, 2916, 3600, 3844, 4624, 5184, 5476, 6400, 7056, 8100, 9604, 10404, 10816, 11664, 12100, 12996, 16384, 17424, 19044, 19600, 22500, 23104, 24964, 26896, 28224, 30276, 32400, 33124, 36864

Offset: 1

Giovanni Teofilatto, Sep 18 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008864, A065472, A065486.

[(p+1)^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

Table[(Prime[n] + 1)^2, {n, 200}] (* Vincenzo Librandi, Mar 27 2014 *)

from sympy import primerange
print([(p+1)**2 for p in primerange(1, 192)]) # Michael S. Branicky, Sep 16 2021

From Amiram Eldar, Jan 23 2021: (Start)
a(n) = A008864(n)^2.
Product_{n>=1} (1 + 1/a(n)) = A065486.
Product_{n>=1} (1 - 1/a(n)) = A065472. (End)
Sum 1/a(n) = A382554. - R. J. Mathar, Mar 31 2025

Corrected and extended by Ray Chandler, Oct 08 2005

A067009 Continued fraction expansion of Product_{p prime} (1 + 1/(p+1)^2).

Original entry on oeis.org

1, 3, 1, 3, 40, 1, 4, 1, 1, 1, 3, 1, 3, 2, 4, 2, 40, 2, 2, 2, 4, 3, 1, 1, 1, 1, 1, 1, 7, 5, 1, 2, 1, 2, 3, 3, 1, 12, 1, 4, 2, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 4, 1, 2, 1, 6, 2, 4, 1, 26, 4, 5, 6, 1, 17, 7, 1, 1, 7, 4, 1, 17, 7, 1, 3, 1, 1, 62, 1, 2, 3, 2, 36, 3, 1, 1, 1, 5, 1, 12, 12, 1, 4, 4, 1, 2, 22, 1

Offset: 0

N. J. A. Sloane, Nov 30 2002

			1.26655850147152857161454711262964...

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

Cf. A065486 (decimal expansion).

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

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003

Offset changed by Andrew Howroyd, Jul 04 2024

A275258 Toth's partial sum over the number of divisors of the greatest unitary divisor.

Original entry on oeis.org

1, 3, 4, 6, 6, 11, 8, 11, 11, 16, 12, 21, 14, 21, 23, 20, 18, 29, 20, 32, 30, 31, 24, 39, 27, 36, 30, 42, 30, 57, 32, 37, 45, 46, 47, 56, 38, 51, 52, 59, 42, 77, 44, 62, 63, 61, 48, 71, 51, 69, 67, 72, 54, 77, 70, 78, 74, 76, 60, 113, 62, 81, 83

Offset: 1

R. J. Mathar, Jul 21 2016

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009), Article 09.5.2, function S**(n).

Cf. A000005, A065486, A165430.

A275258 := proc(n)
    local a,d ;
    a := 0 ;
    for  d in A077610(n) do
        a := a+A005361(d)*A275257(n/d,d) ;
    end do:
    a ;
end proc:
seq(A275258(n),n=1..80) ;

beta[n_] := Times @@ Transpose[FactorInteger[n]][[2]]; phi[x_, n_] := Sum[Boole[ GCD[k, n] == 1 ], {k, 1, x}]; a[n_] := DivisorSum[n, beta[#] * phi[n/#, #] &, GCD[#, n/#] == 1 &]; Array[a, 100] (* Amiram Eldar, Sep 22 2019 *)

a(n) = Sum_{k=1..n} A000005( A165430(n,k) ).

Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n * log(n)^2), where c = A065486. - Amiram Eldar, Dec 22 2023

cp's OEIS Frontend

A110833 a(n) = (prime(n)+1)^2.

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A067009 Continued fraction expansion of Product_{p prime} (1 + 1/(p+1)^2).

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A275258 Toth's partial sum over the number of divisors of the greatest unitary divisor.

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