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A263927 Least solution x to the equation x' = (x+1)/n, where x' is the arithmetic derivative of x; -1 if no solution exists.

%I A263927 #38 Jul 10 2018 02:55:57
%S A263927 30,15,2,3,119,5,209,7,323,559,527,11,779,13,899,1919,1189,17,2507,19,
%T A263927 1763,42455401,2759,23,5249,2911,3239,23519,3827,29,5207,31
%N A263927 Least solution x to the equation x' = (x+1)/n, where x' is the arithmetic derivative of x; -1 if no solution exists.
%C A263927 For n <= 100 unknown terms are a(33), a(57), a(73), a(88) and a(92) (tested up to 10^8).
%C A263927 All solutions for n = 1 are Giuga numbers. - _Paolo P. Lava_, Jul 06 2018
%C A263927 a(33) and a(57) are greater than 10^12, if they exist. - _Giovanni Resta_, Jul 09 2018
%H A263927 Paolo P. Lava, <a href="/A263927/a263927_1.txt">Values for n<=100</a>
%F A263927 a(n) = n-1 if n-1 is a prime.
%F A263927 a(A014574(n)/2) = A037074(n) if n-1 is not a prime. - _Paolo P. Lava_, Jul 06 2018
%e A263927 30' = 31 and 31 = (30 + 1) / 1;
%e A263927 15' = 8 and 8 =(15 + 1) / 2;
%e A263927 2' = 1 and 1 = (2 + 1) / 3.
%p A263927 with(numtheory): P:=proc(q) local a,k,n,p;
%p A263927 for k from 1 to q do for n from 1 to q do
%p A263927 if n*add(op(2,p)/op(1,p),p=ifactors(n)[2])=(n+1)/k then
%p A263927 print(n); break; fi; od; od; end: P(10^9);
%t A263927 f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Table[k = 0; While[f@ k != (k + 1)/n, k++]; k, {n, 21}] (* _Michael De Vlieger_, Nov 05 2015, after _Michael Somos_ at A003415 *)
%o A263927 (PARI) d(x) = {local(fac); if(x<1, 0, fac=factor(x); sum(i=1, matsize(fac)[1], x*fac[i, 2]/fac[i, 1]))}
%o A263927 a(n) = {x=2; while(k, if(d(x) == (x+1)/n, return(x)); x++)} \\ _Altug Alkan_, Nov 05 2015
%Y A263927 Cf. A003415, A007850 (Giuga numbers), A014574, A037074.
%K A263927 nonn,more
%O A263927 1,1
%A A263927 _Paolo P. Lava_, Oct 30 2015
%E A263927 Added a(52) and a(58) in a-file by _Giovanni Resta_, Jul 09 2018