A255354 a(n) = smallest number k such that (k + n)' = k', or -1 if no such number exists, where k' is the arithmetic derivative of k.

2, 3, 2, 3, 2, 5, 110, 3, 2, 3, 2, 5, 50145, 3, 2, 3, 2, 5, 53115, 3, 2, 7, 189, 5, 273, 3, 2, 3, 2, 7, 75, 5, 930642191642, 3, 2, 5, 165, 3, 2, 3, 2, 5, 12, 3, 2, 7, 99, 5, 182, 3, 2, 7, 706, 5, 1523965807, 3, 2, 3, 2, 7, 494, 5

Offset

1,1

Comments

The sequence begins (first 100 terms):

2, 3, 2, 3, 2, 5, 110, 3, 2, 3, 2, 5, 50145, 3, 2, 3, 2, 5, 53115, 3, 2, 7, 189, 5, 273, 3, 2, 3, 2, 7, 75, 5, 930642191642, 3, 2, 5, 165, 3, 2, 3, 2, 5, 12, 3, 2, 7, 99, 5, 182, 3, 2, 7, 706, 5, 1523965807, 3, 2, 3, 2, 7, 494, 5, -1, 3, 2, 5, 1151559, 3, 2, 3, 2, 7, 705, 5, 20, 3, 2, 5, 4526, 3, 2, 7, 1102, 5, 1509626, 3, 2, 13, 778, 7, 226429394, 5, -1, 3, 2, 5, 1910, 3, 2, 3 where the other missing terms (designated by -1: a(63), a(93)) are > 10^12, if they exist).

a(91) = 226429394. - Michel Marcus, Feb 28 2015

a(63), a(93) > 10^12. - Giovanni Resta, Jun 22 2018

Links

Table of n, a(n) for n=1..62.

Example

a(1) = 2 because (2 + 1)' = 2' = 1.
a(2) = 3 because (3 + 2)' = 3' = 1.
a(3) = 2 because (2 + 3)' = 2' = 1.
...
a(7) = 110 because (110 + 7)' = 110' = . Etc.

Maple

with(numtheory): P:=proc(q) local a, b, k, n, p;
for n from 1 to q do for k from 1 to q do
a:=k*add(op(2, p)/op(1, p), p=ifactors(k)[2]); b:=(k+n)*add(op(2, p)/op(1, p), p=ifactors(k+n)[2]);
if a=b then print(k); break; fi; od;
od; end: P(10^20);

Crossrefs

Cf. A003415, A007015, A007365, A015886, A065559.

Sequence in context: A087317 A086489 A015886 * A344349 A318620 A287748

Adjacent sequences: A255351 A255352 A255353 * A255355 A255356 A255357

Keyword

nonn,more

Author

Paolo P. Lava, Feb 24 2015

Extensions

a(33)-a(62) from Giovanni Resta, Jun 22 2018

Status

approved