A210935 Numbers n for which n*n'/(n+n') is an integer, where n' is the arithmetic derivative of n.

1, 4, 64, 80, 108, 270, 351, 432, 729, 768, 864, 2916, 5184, 5832, 6250, 6912, 12096, 13500, 16384, 25600, 32832, 34992, 37500, 39366, 43200, 46656, 50000, 73008, 74304, 81648, 84375, 110592, 131250, 138240, 143748, 153664, 172800, 176418, 200000, 225000

Offset: 1

Author

Paolo P. Lava, May 11 2012

Keywords

• nonn

Comments

Only eleven odd numbers in the first 150 terms: a(1)=1, a(7)=351, a(9)=729, a(31)=84375, a(76)=2470629, a(78)=2709375, a(87)=4159375, a(89)=4348377, a(115)=13286025, a(126)=22235661, a(128)=25059375.

Examples

			n=729; n'=1458; n*n'/(n+n')=486.
		

Links

• Paolo P. Lava and Donovan Johnson, Table of n, a(n) for n = 1..500 (first 150 terms from Paolo P. Lava)

Crossrefs

Cf. A003415, A165562.

Programs

Maple

with(numtheory);
A210935:= proc(n)
local a,i,p,pfs;
for i from 1 to n do
    pfs:=ifactors(i)[2];  a:=i*add(op(2,p)/op(1,p),p=pfs) ;
    if a*i/(a+i)=trunc(a*i/(a+i)) then print(i); fi;
od; end:
A210935(100000000);