%I #32 Dec 20 2022 16:00:35

%S 18,1554,53922,726,1218,3230934,966,798,546,462,210,330,390,510,690,

%T 870,930,1110,1230,1290,748146,1590,1770,2010,2130,2190,10010,2370,

%U 2490,2910,3030,3210,3270,325374,16288662,3810

%N Least solution of the equation x' = x + p for increasing values of p, where x' is the arithmetic derivative of x and p is an odd prime. a(n)= -1 if no such solution exists.

%C First possible values without solutions are a(130), a(311), a(328), a(403) ...

%C If they exist, a(38), a(51) > 7*10^11. a(40) = 53086075098, a(43) = 2205467286, a(59) = 1343266518. - _Giovanni Resta_, Jun 08 2015

%C From _Hiroaki Yamanouchi_, Aug 26 2015: (Start)

%C a(130) > 5*10^13.

%C a(38) <= 6757220092414242, a(82) <= 60382338056274, a(86) <= 4904019996019293138, a(100) <= 20288582235746170309362, a(103) <= 103153074623970054, a(185) <= 359626712451821152206, a(236) <= 41507154995420198557374 and a(378) <= 19746050339765838.

%C a(51) = 49441344782046, a(70) = 487059258, a(88) = 701726466, a(91) = 2192479278, a(116) = 1835382005358, a(126) = 64331242458, a(133) = 2179114806, a(141) = 23383324866 and a(142) = 2175820746. (End)

%H Paolo P. Lava, <a href="/A258336/a258336_2.txt">Solutions for the first 100 odd primes (-1 if no solution has been detected)</a>

%e a(2) = 18 because the second prime is 3 and the solutions of the equation x' = x + 3 are 18, 315, ... where 18 is the least one.

%e a(3) = 1554 because the third prime is 5 and the solutions of the equation x' = x + 5 are 1554, 27950, ... where 1554 is the least one.

%p with(numtheory); P:=proc(q) local b,c,n,p;

%p for c from 2 to q do b:=ithprime(c); for n from 1 to q do

%p if n+b=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]) then print(n);

%p break; fi; od; od; end: P(10^9);

%o (PARI) der(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);

%o a(n) = {k=1; p=prime(n); while(der(k) != k+p, k++); k;} \\ _Michel Marcus_, Aug 28 2015

%Y Cf. A000040, A003415.

%K nonn,more

%O 2,1

%A _Paolo P. Lava_, May 27 2015