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

1,1

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

Links

Table of n, a(n) for n=1..32.

Paolo P. Lava, Values for n<=100

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A040874 A147454 A147077 * A033350 A070293 A194709

Adjacent sequences: A263924 A263925 A263926 * A263928 A263929 A263930

Keyword

nonn,more

Author

Paolo P. Lava, Oct 30 2015

Extensions

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

Status

approved