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

1,3

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'.

Links

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

Example

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.

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);

Crossrefs

Cf. A003415, A046311, A165558.

Sequence in context: A341887 A031465 A045081 * A235060 A260243 A206137

Adjacent sequences: A294150 A294151 A294152 * A294154 A294155 A294156

Keyword

nonn

Author

Paolo P. Lava, Oct 24 2017

Status

approved