cp's OEIS Frontend

A083502 Smallest k such that n*(n+k) + 1 is an n-th power.

%I A083502 #15 Aug 03 2025 16:10:53
%S A083502 1,2,18,16,1550,2598,299586,812,29118,348678430,67546215506,20345040,
%T A083502 61054982557998,281241170407078,76861433640456450,2690404,
%U A083502 128583032925805678334,211927625850,275941052631578947368402,174339200
%N A083502 Smallest k such that n*(n+k) + 1 is an n-th power.
%C A083502 Sequence is obviously infinite.
%C A083502 If the sequence is restricted to only prime n's, the sequence increases absolutely. See comment in A083503.
%C A083502 [Since there is actually no comment in A083503: this probably means to say that (conjectural!) A083503(prime(n)) = A008864(n) which leads to a(p) = Sum_{s=2..p} binomial(p,s)*p^(s-1) for primes p, an increasing subsequence. - _R. J. Mathar_, Aug 01 2025]
%C A083502 a(n) = (x^n-1)/n - n, where x is the least integer > 1 with x^n == 1 (mod n). - _Robert Israel_, Aug 01 2025
%H A083502 Robert Israel, <a href="/A083502/b083502.txt">Table of n, a(n) for n = 1..388</a>
%p A083502 A083502 := proc(n)
%p A083502     local a,b ;
%p A083502     if n = 1 then
%p A083502         1 ;
%p A083502     else
%p A083502         for b from 2 do
%p A083502             a := (b^n-1)/n-n ;
%p A083502             if type( a,'integer') then
%p A083502                 return  a;
%p A083502             end if;
%p A083502         end do:
%p A083502     end if;
%p A083502 end proc:
%p A083502 seq(A083502(n),n=1..20) ; # _R. J. Mathar_, Aug 01 2025
%p A083502 # alternative
%p A083502 f:= proc(n) local X,S;
%p A083502   S:= min(map(t -> subs(t,X), {msolve(X^n = 1, n)} minus {{X=1}}));
%p A083502   if S = infinity then ((n+1)^n - 1)/n - n else (S^n-1)/n - n fi
%p A083502 end proc:
%p A083502 f(1):= 1:
%p A083502 map(f, [$1..50]); # _Robert Israel_, Aug 01 2025
%t A083502 Do[i = 2; While[k = (i^n - 1)/n - n; !IntegerQ[k], i++ ]; Print[k], {n, 1, 20}]
%Y A083502 The i's in the above Mathematica coding, except for a(1), give A055670.
%Y A083502 Cf. A083500, A083503.
%K A083502 nonn
%O A083502 1,2
%A A083502 _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003
%E A083502 Edited and extended by _Robert G. Wilson v_, May 11 2003