cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

A323046 Numbers that are neither 3-smooth nor a sum of two 3-smooth numbers.

Original entry on oeis.org

23, 46, 47, 53, 61, 69, 71, 77, 79, 92, 94, 95, 101, 103, 106, 107, 115, 119, 121, 122, 125, 127, 133, 138, 139, 141, 142, 143, 149, 151, 154, 157, 158, 159, 161, 167, 169, 173, 175, 179, 181, 183, 184, 185, 187, 188, 190, 191, 197, 199, 202, 203, 205, 206, 207, 211, 212, 213, 214, 215, 221, 223, 227, 229, 230, 231, 233
Offset: 1

Views

Author

Carlos Alves, Jan 03 2019

Keywords

Examples

			23 is not in A003586; also 22 (23-1), 21 (23-2), 20 (23-3), 19 (23-2*2), 17 (23-2*3), 15 (23-2*2*2), 14 (23-3*3), 11 (23-2*2*3), 7 (23-2*2*2*2), 5 (23-2*3*3) are not in A003586.

Links

Crossrefs

Cf. A003586, A237442, A323047, A323049, A323050. Subsequence of A081329.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
S:= {seq(seq(2^i*3^j,i=0..ilog2(N/3^j)),j=0..floor(log[3](N)))}:
sort(convert({$1..N} minus S minus map(t -> op(map(`+`, S,t)), S), list)); # Robert Israel, May 19 2019

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

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

A081327 Smallest 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, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 0, 6, 1, 2, 3, 1, 2, 3, 4, 8, 1, 2, 3, 4, 1, 2, 3, 4, 9, 6, 16, 8, 9, 0, 0, 12, 1, 2, 3, 4, 0, 6, 1, 2, 3, 4, 27, 6, 0, 8, 9, 16, 1, 2, 3, 4, 0, 6, 0, 8, 1, 2, 3, 4, 0, 6, 0, 8, 9, 1, 2, 3, 4, 32, 6, 16, 8, 9, 27, 0, 12, 0, 0, 24, 1, 2, 3, 4, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Examples

			a(32) = 8 = 2^3, as 32 = 8 + 3*2^3.

Crossrefs

Cf. A081329, A081330, A081328, A000040, A003586.

Programs

  • Mathematica
    smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
a[n_] := Module[{m}, For[m = 1, mJean-François Alcover, Oct 14 2021 *)

Formula

a(n) = 0 iff A081326(n) = 0.
If a(n) > 0: a(n)+A081328(n) = n.

A081331 Number of numbers <= n that can be written as sum of two 3-smooth numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 43, 43, 44, 45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62, 62, 63, 63, 64, 65
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Links

Crossrefs

Cf. A081330, A081326, A003586, A081329.

Programs

  • Mathematica
    With[{max = 100}, sm = Select[Range[max], # == Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]) &]; Accumulate@ Table[Boole[IntegerPartitions[n, {2}, sm] != {}], {n, 1, max}]] (* Amiram Eldar, Mar 26 2025 *)
Showing 1-6 of 6 results.

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