A067371 Arithmetic derivatives of 3-smooth numbers.

0, 1, 1, 4, 5, 12, 6, 16, 32, 21, 44, 27, 80, 60, 112, 81, 192, 156, 108, 272, 216, 448, 384, 297, 640, 540, 405, 1024, 912, 756, 1472, 1296, 1053, 2304, 2112, 1836, 1458, 3328, 3024, 2592, 5120, 4800, 4320, 3645, 7424, 6912, 6156, 11264, 5103, 10752

Offset: 1

Author

Reinhard Zumkeller, Mar 20 2002, revised: Jul 19 2003

Keywords

• nonn

Examples

			a(18) = A003415(A003586(18)) = A003415(72) = A003415(2^3*3^2) = (3*3+2*2)*2^(3-1)*3^(2-1) = (9+4)*2^2*3^1 = 13*4*3 = 156.
a(27) = A003415(A003586(27)) = A003415(243) = A003415(2^0*3^5) = (3*0+2*5)*2^(0-1)*3^(5-1) = ((0+10)/2)*3^4 = 5*81 = 405.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001787, A003415, A003586, A027471, A022328, A022329.

Programs

Mathematica

s = {}; m = 12; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; ad[1] = 0; ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); ad /@ Union[s] (* Amiram Eldar, Jan 29 2020 *)

Formula

A003415(2^i+3^j) = (3*i + 2*j) * 2^(i-1) * 3^(j-1), i, j >=0.

a(n) = A003415(A003586(n)).