A076943 Smallest k > 0 such that n*k + 1 is an n-th power.

1, 4, 21, 20, 1555, 2604, 299593, 820, 29127, 348678440, 67546215517, 20345052, 61054982558011, 281241170407092, 76861433640456465, 2690420, 128583032925805678351, 211927625868, 275941052631578947368421, 174339220

Offset: 1

Author

Amarnath Murthy, Oct 19 2002

Keywords

• nice
• nonn

Examples

			For n = 7, 1 + 7*a(7) = 1 + 7*299593 = 2097152 = 2^21 = 8^7.
For n = 10, 1 + 10*a(10) = 1 + 10*348678440 = 3486784401 = 3^20 = 9^10. - _Marius A. Burtea_, Jun 01 2019
		

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A076942, A076944, A074792.

Programs

Magma

sol:=[];
for u in [1..20] do
   for k in [2..100] do
      if IsIntegral((k^u-1)/u) then sol[u]:=(k^u-1)/u; break; end if;
   end for;
end for;
sol; // Marius A. Burtea, Jun 01 2019

Mathematica

Do[k = 2; While[ !IntegerQ[(k^n - 1)/n], k++ ]; Print[(k^n - 1)/n], {n, 1, 20}] (* Robert G. Wilson v, Oct 21 2002 *)

Formula

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

a(p^k) = ((p+1)^(p^k) - 1) / p^k. - Charlie Neder, May 23 2019

a(2*p) = ((2*p-1)^(2*p) - 1) / (2*p). - Charlie Neder, May 23 2019

Extensions

Edited, corrected and extended by Robert G. Wilson v, Oct 21 2002