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 1-7 of 7 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 *)

A081329 Numbers having no representation as sum of two 3-smooth numbers.

Original entry on oeis.org

1, 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
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Comments

Complement of A081330.

Examples

			For all 3-smooth numbers m<23 the greatest prime factor of 23-m is >3: 23-1=2*11, 23-2=3*7, 23-3=4*5, 23-2^2=19, 23-2*3=17, 23-2^3=3*5, 23-3^2=2*7, 23-3*2^2=11, 23-2^4=7, 23-2*3^2=5, therefore 23 is a term.

Links

Crossrefs

Cf. A003586, A081326, A081330.

Programs

  • Mathematica
    max = 300; Complement[Range[max], 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 *)

Formula

A081326(a(n)) = 0.

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.

A081333 Numbers having more than one partition into two 3-smooth numbers.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 70, 72, 73, 75, 76, 78, 80, 81, 82, 84, 88, 90, 96, 97, 99, 100, 102, 104, 105, 108, 112
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Links

Crossrefs

Cf. A003586, A081326, A081332, A081330.

Formula

A081326(a(n)) > 1.

A081332 Numbers having a unique partition into two 3-smooth numbers.

Original entry on oeis.org

2, 3, 29, 31, 37, 41, 43, 49, 55, 58, 59, 62, 65, 67, 74, 83, 85, 86, 87, 89, 91, 93, 98, 109, 110, 111, 113, 116, 118, 123, 124, 130, 131, 134, 137, 147, 148, 155, 163, 165, 166, 170, 172, 174, 177, 178, 182, 186, 193, 195, 196, 201, 209, 217, 218, 220, 222, 226
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 19 2003

Keywords

Crossrefs

Cf. A003586, A081326, A081333, A081330.

Programs

  • Mathematica
    nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
P[n_] := IntegerPartitions[n, {2}, TakeWhile[S, # < n &]];
selQ[n_] := Length[P[n]] == 1;
Select[Range[nmax], selQ] (* Jean-François Alcover, Oct 14 2021 *)

Formula

A081326(a(n)) = 1.

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 1-7 of 7 results.

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