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

%I A125598 #42 Aug 15 2022 04:24:50
%S A125598 0,1,5,31,259,2801,37449,597871,11111111,235794769,5628851293,
%T A125598 149346699503,4361070182715,139013933454241,4803839602528529,
%U A125598 178901440719363487,7143501829211426575,304465936543600121441
%N A125598 a(n) = ((n+1)^(n-1) - 1)/n.
%C A125598 Odd prime p divides a(p-2).
%C A125598 a(n) is prime for n = {3,4,6,74, ...}; prime terms are {5, 31, 2801, ...}.
%C A125598 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
%H A125598 G. C. Greubel, <a href="/A125598/b125598.txt">Table of n, a(n) for n = 1..350</a>
%F A125598 a(n) = ((n+1)^(n-1) - 1)/n.
%F A125598 a(n) = (A000272(n+1) - 1)/n.
%F A125598 a(2k-1)/(2k+1) = A125599(k) for k>0.
%F A125598 From _Mathew Englander_, Dec 17 2020: (Start)
%F A125598 a(n) = (A060072(n+1) - A083069(n-1))/2.
%F A125598 For n > 1, a(n) = Sum_{k=0..n-2} (n+1)^k.
%F A125598 For n > 1, a(n) = Sum_{j=0..n-2} n^j*C(n-1,j+1). (End)
%t A125598 Table[((n+1)^(n-1)-1)/n, {n,25}]
%o A125598 (Sage) [gaussian_binomial(n,1,n+2) for n in range(0,18)] # _Zerinvary Lajos_, May 31 2009
%o A125598 (Magma) [((n+1)^(n-1) -1)/n: n in [1..25]]; // _G. C. Greubel_, Aug 15 2022
%Y A125598 Cf. A000272 (n^(n-2)), A125599.
%Y A125598 Cf. other sequences of generalized repunits, such as A125118, A053696, A055129, A060072, A031973, A173468, A023037, A119598, A085104, and A162861.
%K A125598 nonn
%O A125598 1,3
%A A125598 _Alexander Adamchuk_, Nov 26 2006