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 A294153 Numbers k = a * b, such that k' = a' * b' where k', a' and b' are the arithmetic derivatives of k, a and b. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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])); b:=(n/j[k])*add(op(2, p)/op(1, p), p=ifactors(n/j[k])); c:=n*add(op(2, p)/op(1, p), p=ifactors(n)); 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

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Last modified March 24 11:22 EDT 2023. Contains 361479 sequences. (Running on oeis4.)