A129290 -id:A129290 - cp's OEIS frontend

A056852 a(n) = (p^p + 1)/(p + 1), where p = prime(n).

7, 521, 102943, 23775972551, 21633936185161, 45957792327018709121, 98920982783015679456199, 870019499993663001431459704607, 85589538438707037818727607157700537549449, 533411691585101123706582594658103586126397951, 277766709362573247738903423315679814371773581141321037961

Offset: 2

Robert G. Wilson v, Aug 30 2000

nonn

From Lorenzo Sauras Altuzarra, Nov 27 2022: (Start)
Are all terms pairwise coprime? If so, they induce a permutation of the natural numbers, as Fermat numbers do (see A343767).
Are all terms squarefree?
A342173(n) <= length(a(n)) = A055642(a(n)) (the proof is due to Jinyuan Wang). (End)

T. D. Noe, Table of n, a(n) for n = 2..26
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).

Cf. A001039, A055642, A129290, A342173, A343767.

a := n -> (ithprime(n)^ithprime(n)+1)/(ithprime(n)+1): # Lorenzo Sauras Altuzarra, Nov 27 2022

Table[ (Prime[ n ]^Prime[ n ] + 1)/(Prime[ n ] + 1), {n, 2, 11} ]
(#^#+1)/(#+1)&/@Prime[Range[2,20]] (* Harvey P. Dale, Apr 23 2015 *)

From Lorenzo Sauras Altuzarra, Nov 27 2022: (Start)
a(n) = Sum_{k=0..prime(n)-1} (-prime(n))^k.
a(n) = F(prime(n), 1)/F(prime(n), 0), where F(b, n) = b^b^n + 1 (i.e., F(b, n) is the n-th base-b Fermat number, see A129290). (End)

A113913 Number of distinct prime factors of 3^(3^n)+1.

Original entry on oeis.org

1, 2, 4, 6, 11, 20, 27

Offset: 0

Jorge Coveiro, Jan 29 2006

The term omega(3^(3^6)+1) was obtained with ECM factoring.

Table[PrimeNu[3^(3^n) + 1], {n,1,10}] (* G. C. Greubel, May 16 2017 *)

for(x=1,6,print1( omega(3^(3^x)+1),","))

a(n) = A001221(A129290(n)).

A129291 a(n) = 1 - 3^(3^n) + 9^(3^n).

Original entry on oeis.org

7, 703, 387400807, 58149737003032434092905183, 196627050475552913618075908526912116282660024455971729157367165907347241304007

Offset: 0

Alexander Adamchuk, Apr 08 2007

nonn

a(n) is the ratio of two consecutive base-3 Fermat numbers A129290(n) = 3^(3^n) + 1 = {4, 28, 19684, 7625597484988, ...}.

Cf. A129290 (3^(3^n) + 1).
Cf. A055777 (3^(3^n)).
Cf. A002061 (central polygonal numbers: n^2 - n + 1).

Mathematica
```
Table[1 - 3^3^n + 9^3^n, {n,0,5}]
```

a(n) = A002061(3^(3^n)). a(n) = A129290(n+1) / A129290(n).

cp's OEIS Frontend

A056852 a(n) = (p^p + 1)/(p + 1), where p = prime(n).

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A113913 Number of distinct prime factors of 3^(3^n)+1.

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A129291 a(n) = 1 - 3^(3^n) + 9^(3^n).

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