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

%I A076943 #27 Feb 16 2022 23:10:20
%S A076943 1,4,21,20,1555,2604,299593,820,29127,348678440,67546215517,20345052,
%T A076943 61054982558011,281241170407092,76861433640456465,2690420,
%U A076943 128583032925805678351,211927625868,275941052631578947368421,174339220
%N A076943 Smallest k > 0 such that n*k + 1 is an n-th power.
%H A076943 Marius A. Burtea, <a href="/A076943/b076943.txt">Table of n, a(n) for n = 1..150</a>
%F A076943 a(n) <= ((n+1)^n - 1) / n.
%F A076943 a(p^k) = ((p+1)^(p^k) - 1) / p^k. - _Charlie Neder_, May 23 2019
%F A076943 a(2*p) = ((2*p-1)^(2*p) - 1) / (2*p). - _Charlie Neder_, May 23 2019
%e A076943 For n = 7, 1 + 7*a(7) = 1 + 7*299593 = 2097152 = 2^21 = 8^7.
%e A076943 For n = 10, 1 + 10*a(10) = 1 + 10*348678440 = 3486784401 = 3^20 = 9^10. - _Marius A. Burtea_, Jun 01 2019
%t A076943 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 *)
%o A076943 (Magma) sol:=[];
%o A076943 for u in [1..20] do
%o A076943    for k in [2..100] do
%o A076943       if IsIntegral((k^u-1)/u) then sol[u]:=(k^u-1)/u; break; end if;
%o A076943    end for;
%o A076943 end for;
%o A076943 sol; // _Marius A. Burtea_, Jun 01 2019
%Y A076943 Cf. A076942, A076944, A074792.
%K A076943 nice,nonn
%O A076943 1,2
%A A076943 _Amarnath Murthy_, Oct 19 2002
%E A076943 Edited, corrected and extended by _Robert G. Wilson v_, Oct 21 2002