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%I #22 Oct 15 2021 07:29:33
%S 0,1,1,1,2,2,2,3,3,4,5,5,6,7,9,10,11,13,14,17,19,22,25,28,33,37,41,43,
%T 49,55,65,73,82,86,97,109,122,129,145,163,171,193,217,244,257,289,325,
%U 342,365,385,433,487,513,577,649,683,730,769,865,973,1025,1094,1153
%N Number of partitions of the n-th 3-smooth number into parts 2 and 3.
%H Michael De Vlieger, <a href="/A090184/b090184.txt">Table of n, a(n) for n = 1..10000</a>
%F a(2^i * 3^j) = floor(2^(i-1) * 3^(j-1) + 1), i*j>0.
%F a(n) = A103221(A003586(n)).
%e n=11: A003586(11) = 2^3 * 3 = 24: 3+3+3+3+3+3+3+3 = 3+3+3+3+3+3+2+2+2 = 3+3+3+3+2+2+2+2+2+2 = 3+3+2+2+2+2+2+2+2+2+2 = 2+2+2+2+2+2+2+2+2+2+2+2: a(11)=5.
%t smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
%t Length[IntegerPartitions[#, All, {2, 3}]]& /@ Select[Range[10000], smooth3Q] (* _Jean-François Alcover_, Oct 13 2021 *)
%t With[{nn = 6^5}, Map[Floor[#/2] - Floor[#/3] &, Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2[#]}, {b, 0, Log[3, #/(2^a)]}] &[nn] + 2]] (* _Michael De Vlieger_, Oct 13 2021 *)
%Y Cf. A022328, A022329.
%Y Cf. A003586, A008615.
%Y Cf. A103221, A117222, A117220, A117221.
%K nonn
%O 1,5
%A _Reinhard Zumkeller_, Jan 21 2004
%E Offset changed to 1 by _Alois P. Heinz_, Oct 15 2021
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