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

30, 15, 2, 3, 119, 5, 209, 7, 323, 559, 527, 11, 779, 13, 899, 1919, 1189, 17, 2507, 19, 1763, 42455401, 2759, 23, 5249, 2911, 3239, 23519, 3827, 29, 5207, 31

Offset: 1

Author

Paolo P. Lava, Oct 30 2015

Keywords

• nonn
• more

Comments

For n <= 100 unknown terms are a(33), a(57), a(73), a(88) and a(92) (tested up to 10^8).

All solutions for n = 1 are Giuga numbers. - Paolo P. Lava, Jul 06 2018

a(33) and a(57) are greater than 10^12, if they exist. - Giovanni Resta, Jul 09 2018

Examples

			30' = 31 and 31 = (30 + 1) / 1;
15' = 8 and 8 =(15 + 1) / 2;
2' = 1 and 1 = (2 + 1) / 3.
		

Links

• Paolo P. Lava, Values for n<=100

Crossrefs

Cf. A003415, A007850 (Giuga numbers), A014574, A037074.

Programs

Maple

with(numtheory): P:=proc(q) local a,k,n,p;
for k from 1 to q do for n from 1 to q do
if n*add(op(2,p)/op(1,p),p=ifactors(n)[2])=(n+1)/k then
print(n); break; fi; od; od; end: P(10^9);

Mathematica

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 *)

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]))}
a(n) = {x=2; while(k, if(d(x) == (x+1)/n, return(x)); x++)} \\ Altug Alkan, Nov 05 2015

Formula

a(n) = n-1 if n-1 is a prime.

a(A014574(n)/2) = A037074(n) if n-1 is not a prime. - Paolo P. Lava, Jul 06 2018

Extensions

Added a(52) and a(58) in a-file by Giovanni Resta, Jul 09 2018