A125598 a(n) = ((n+1)^(n-1) - 1)/n.

0, 1, 5, 31, 259, 2801, 37449, 597871, 11111111, 235794769, 5628851293, 149346699503, 4361070182715, 139013933454241, 4803839602528529, 178901440719363487, 7143501829211426575, 304465936543600121441

Offset: 1

Author

Alexander Adamchuk, Nov 26 2006

Keywords

• nonn

Comments

Odd prime p divides a(p-2).

a(n) is prime for n = {3,4,6,74, ...}; prime terms are {5, 31, 2801, ...}.

a(n) is the (n-1)-th generalized repunit in base (n+1). For example, a(5) = 259 which is 1111 in base 6. - Mathew Englander, Oct 20 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A000272 (n^(n-2)), A125599.

Cf. other sequences of generalized repunits, such as A125118, A053696, A055129, A060072, A031973, A173468, A023037, A119598, A085104, and A162861.

Programs

Magma

[((n+1)^(n-1) -1)/n: n in [1..25]]; // G. C. Greubel, Aug 15 2022

Mathematica

Table[((n+1)^(n-1)-1)/n, {n,25}]

Sage

[gaussian_binomial(n,1,n+2) for n in range(0,18)] # Zerinvary Lajos, May 31 2009

Formula

a(n) = ((n+1)^(n-1) - 1)/n.

a(n) = (A000272(n+1) - 1)/n.

a(2k-1)/(2k+1) = A125599(k) for k>0.

From Mathew Englander, Dec 17 2020: (Start)

a(n) = (A060072(n+1) - A083069(n-1))/2.

For n > 1, a(n) = Sum_{k=0..n-2} (n+1)^k.

For n > 1, a(n) = Sum_{j=0..n-2} n^j*C(n-1,j+1). (End)