cp's OEIS Frontend

A051442 a(n) = n^(n+1)+(n+1)^n.

%I A051442 #35 Sep 08 2022 08:44:59
%S A051442 1,3,17,145,1649,23401,397585,7861953,177264449,4486784401,
%T A051442 125937424601,3881436747409,130291290501553,4731091158953433,
%U A051442 184761021583202849,7721329860319737601,343809097055019694337,16248996011806421522977
%N A051442 a(n) = n^(n+1)+(n+1)^n.
%C A051442 Odd prime p divides a(p-2). For n>1, a(prime(n)-2)/prime(n) = A125074(n) = {1, 29, 3343, 407889491, 298572057493, 454195874136455153, ...}. Prime p divides a((p+5)/2) for p = {19, 23, 61}. - _Alexander Adamchuk_, Nov 18 2006
%C A051442 From _Mathew Englander_, Jul 08 2020: (Start)
%C A051442 For all n != 1, a(n) mod 8 = 1.
%C A051442 If n mod 6 = 0, 3, or 5, then a(n) mod 6 = 1. If n mod 6 = 1, then a(n) mod 6 = 3. If n mod 6 = 2 or 4, then a(n) mod 6 = 5.
%C A051442 For all n, a(n)-1 is a multiple of n^2.
%C A051442 For n odd  and n >= 3, a(n)-1 is a multiple of (n+1)^2.
%C A051442 For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
%C A051442 For proofs, see the Englander link. (End)
%H A051442 Vincenzo Librandi, <a href="/A051442/b051442.txt">Table of n, a(n) for n = 0..200</a>
%H A051442 Mathew Englander, <a href="/A051442/a051442.pdf">Notes on OEIS A051442</a>
%F A051442 a(n) = (n + e + o(1)) * n^n. - _Charles R Greathouse IV_, Jan 12 2012
%F A051442 From _Mathew Englander_, Jul 08 2020: (Start)
%F A051442 a(n) = A093898(n+1, n) for n >= 1.
%F A051442 a(n) = a(n-1) + A258389(n) for n >= 1.
%F A051442 a(n) = A007778(n) + A000169(n+1).
%F A051442 (End)
%t A051442 Table[n^(n+1)+(n+1)^n,{n,0,20}] (* _Harvey P. Dale_, Oct 02 2018 *)
%o A051442 (PARI) a(n)=(n+1)^n+n^(n+1) \\ _Charles R Greathouse IV_, Jan 12 2012
%o A051442 (Magma)[n^(n+1)+(n+1)^n: n in [0..20]]; // _Vincenzo Librandi_, Jan 12 2012
%o A051442 (Maxima) A051442[n]:=n^(n+1)+(n+1)^n$ makelist(A051442[n],n,0,30); /* _Martin Ettl_, Oct 29 2012 */
%Y A051442 Cf. A007925, A051443, A125074, A076980, A173054, A208506.
%K A051442 nonn,easy
%O A051442 0,2
%A A051442 _N. J. A. Sloane_