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

Author

Paolo P. Lava, Feb 24 2015

Keywords

• nonn
• more

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

Examples

			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.
		

Crossrefs

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

Programs

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);

Extensions

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