A294153 Numbers k = a * b, such that k' = a' * b' where k', a' and b' are the arithmetic derivatives of k, a and b.

0, 1, 256, 512, 1152, 1728, 1920, 3072, 3456, 7776, 11664, 12800, 12960, 20736, 23328, 52488, 72000, 78732, 81920, 86400, 87480, 100352, 110208, 124800, 139968, 153216, 157464, 200000, 219520, 263424, 294912, 321024, 336000, 354294, 400000, 486000, 486720, 531441

Offset: 1

Author

Paolo P. Lava, Oct 24 2017

Keywords

• nonn

Comments

A046311 is a subset of this sequence.

Some numbers admit more than one couple of divisors a, b: 3456 = 8 * 432 = 54 * 64 and 3456' = 15552 = 8' * 432' = 12 * 1296 = 54' * 64' = 81 * 192.

Apart from the first term, squares of A165558 are part of the sequence. In A165558 n' = 2 * n and therefore (n^2)' = 2 * n * n' = 2 * n * 2 * n = (2 * n)^2. Thus n^2 = n * n and (n^2)' = n' * n'.

Examples

			a(0) = 0 because 0 = 0 * b and 0' = 0' * b' = 0;
a(1) = 1 because 1 = 1 * 1 and 1' = 1' * 1' = 0;
a(2) = 256 because 256 = 16 * 16 and 256' = 16' * 16' = 32 * 32 = 1024;
a(3) = 512 because 512 = 8 * 64 and 512' = 8' * 64' = 12 * 192 = 2304.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A003415, A046311, A165558.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,j,k,n,p;
for n from 1 to q do j:=sort([op(divisors(n))]);
for k from 2 to trunc((nops(j)+1)/2) do
a:=j[k]*add(op(2,p)/op(1,p), p=ifactors(j[k])[2]);
b:=(n/j[k])*add(op(2,p)/op(1,p), p=ifactors(n/j[k])[2]);
c:=n*add(op(2,p)/op(1,p), p=ifactors(n)[2]);
if c=a*b then print(n); break; fi; od; od; end: P(10^6);