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

1, 2, 18, 16, 1550, 2598, 299586, 812, 29118, 348678430, 67546215506, 20345040, 61054982557998, 281241170407078, 76861433640456450, 2690404, 128583032925805678334, 211927625850, 275941052631578947368402, 174339200

Offset: 1

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

Keywords

• nonn

Comments

Sequence is obviously infinite.

If the sequence is restricted to only prime n's, the sequence increases absolutely. See comment in A083503.

[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]

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

Links

• Robert Israel, Table of n, a(n) for n = 1..388

Crossrefs

The i's in the above Mathematica coding, except for a(1), give A055670.

Cf. A083500, A083503.

Programs

Maple

A083502 := proc(n)
    local a,b ;
    if n = 1 then
        1 ;
    else
        for b from 2 do
            a := (b^n-1)/n-n ;
            if type( a,'integer') then
                return  a;
            end if;
        end do:
    end if;
end proc:
seq(A083502(n),n=1..20) ; # R. J. Mathar, Aug 01 2025
# alternative
f:= proc(n) local X,S;
  S:= min(map(t -> subs(t,X), {msolve(X^n = 1, n)} minus {{X=1}}));
  if S = infinity then ((n+1)^n - 1)/n - n else (S^n-1)/n - n fi
end proc:
f(1):= 1:
map(f, [$1..50]); # Robert Israel, Aug 01 2025

Mathematica

Do[i = 2; While[k = (i^n - 1)/n - n; !IntegerQ[k], i++ ]; Print[k], {n, 1, 20}]

Extensions

Edited and extended by Robert G. Wilson v, May 11 2003