A258336 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.

18, 1554, 53922, 726, 1218, 3230934, 966, 798, 546, 462, 210, 330, 390, 510, 690, 870, 930, 1110, 1230, 1290, 748146, 1590, 1770, 2010, 2130, 2190, 10010, 2370, 2490, 2910, 3030, 3210, 3270, 325374, 16288662, 3810

Offset

2,1

Comments

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

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

From Hiroaki Yamanouchi, Aug 26 2015: (Start)

a(130) > 5*10^13.

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.

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)

Links

Table of n, a(n) for n=2..37.

Paolo P. Lava, Solutions for the first 100 odd primes (-1 if no solution has been detected)

Example

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.
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.

Maple

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

Prog

(PARI) der(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
a(n) = {k=1; p=prime(n); while(der(k) != k+p, k++); k; } \\ Michel Marcus, Aug 28 2015

Crossrefs

Cf. A000040, A003415.

Sequence in context: A160252 A276015 A210823 * A211310 A160307 A003030

Adjacent sequences: A258333 A258334 A258335 * A258337 A258338 A258339

Keyword

nonn,more

Author

Paolo P. Lava, May 27 2015

Status

approved