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A294153
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Numbers k = a * b, such that k' = a' * b' where k', a' and b' are the arithmetic derivatives of k, a and b.
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1
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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
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OFFSET
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1,3
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COMMENTS
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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'.
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LINKS
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Paolo P. Lava, Table of n, a(n) for n = 1..100
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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Cf. A003415, A046311, A165558.
Sequence in context: A341887 A031465 A045081 * A235060 A260243 A206137
Adjacent sequences: A294150 A294151 A294152 * A294154 A294155 A294156
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KEYWORD
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nonn
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AUTHOR
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Paolo P. Lava, Oct 24 2017
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STATUS
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approved
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