A081327 -id:A081327 - cp's OEIS frontend

A081326 Number of partitions of n into two 3-smooth numbers.

0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 0, 3, 2, 2, 2, 3, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 2, 3, 1, 2, 1, 2, 2, 0, 0, 3, 1, 2, 2, 2, 0, 3, 1, 3, 2, 1, 1, 3, 0, 1, 2, 2, 1, 3, 1, 2, 0, 2, 0, 4, 2, 1, 2, 2, 0, 2, 0, 3, 2, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 0, 1, 0, 0, 3, 2, 1, 3, 2, 0, 2, 0, 2, 2

Offset: 1

Reinhard Zumkeller, Mar 19 2003

nonn

			n=10 has a(10)=3 partitions into 3-smooth numbers: 10=1+3^2=2+2^3=2^2+2*3; n=9 has a(9)=2 partitions into 3-smooth numbers: 9=1+2^3=3+2*3.

Jean-François Alcover, Table of n, a(n) for n = 1..10000
Ivars Peterson, Medieval Harmony.
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A081329, A081330, A081327, A081328, A003586.

nmax = 10000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
P[n_] := IntegerPartitions[n, {2}, TakeWhile[S, # < n &] ];
a[n_] := P[n] // Length;
Array[a, nmax] (* Jean-François Alcover, Oct 13 2021 *)

A081330 Numbers that can be written as sum of two 3-smooth numbers.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82

Offset: 1

Reinhard Zumkeller, Mar 19 2003

nonn

A081326(a(n)) > 0; complement of A081329.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ivars Peterson, Medieval Harmony. [Broken link]
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A003586, A081332, A081333, A081327, A081328.

max = 100; Table[2^i *3^j + 2^k* 3^l, {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[3, max] // Ceiling}, {k, 0, Log[2, max] // Ceiling}, {l, 0, Log[3, max] // Ceiling}] // Flatten // Union // Select[#, # <= max &] & (* Jean-François Alcover, Sep 10 2017 *)

list(lim)=my(v=List(), u=v, N); for(n=0, log(lim\1+.5)\log(3), N=3^n; while(N<=lim, listput(v, N); N<<=1)); v=Set(v); for(i=1,#v,for(j=i,#v, N=v[i]+v[j]; if(N>lim, break); listput(u,N))); Set(u) \\ Charles R Greathouse IV, Sep 07 2014

a(n) >> exp(n^(1/4)). - Charles R Greathouse IV, Aug 22 2011

A081328 Greatest 3-smooth number m such that n-m is also 3-smooth, a(n)=0 if no such 3-smooth number exists.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 6, 6, 8, 9, 9, 9, 12, 12, 12, 12, 16, 16, 18, 18, 18, 18, 0, 18, 24, 24, 24, 27, 27, 27, 27, 24, 32, 32, 32, 32, 36, 36, 36, 36, 32, 36, 27, 36, 36, 0, 0, 36, 48, 48, 48, 48, 0, 48, 54, 54, 54, 54, 32, 54, 0, 54, 54, 48, 64, 64, 64, 64, 0, 64, 0, 64, 72, 72, 72, 72, 0

Offset: 1

Reinhard Zumkeller, Mar 19 2003

nonn

			a(32) = 24 = 3*2^3, as 32 = 2^3 + 24.

Cf. A081329, A081330, A081327, A000040, A003586.

smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
a[n_] := Module[{m}, For[m = n, m >= 1, m--, If[smooth3Q[m], If[smooth3Q[n - m], Return[m]]]]; 0];
Array[a, 77] (* Jean-François Alcover, Oct 17 2021 *)

a(n)=0 iff A081326(n)=0; if a(n)>0: a(n)+A081327(n)=n.

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A081326 Number of partitions of n into two 3-smooth numbers.

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A081330 Numbers that can be written as sum of two 3-smooth numbers.

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A081328 Greatest 3-smooth number m such that n-m is also 3-smooth, a(n)=0 if no such 3-smooth number exists.

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