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.

A212214 Number of representations of n as a sum of products of pairs of positive integers, n = Sum_{k=1..m} i_k*j_k with i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_k*j_k<=i_{k+1}*j_{k+1}.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 18, 29, 39, 57, 74, 109, 138, 192, 247, 335, 421, 565, 703, 926, 1151, 1484, 1828, 2349, 2868, 3624, 4423, 5538, 6706, 8345, 10048, 12394, 14895, 18219, 21789, 26549, 31596, 38226, 45415, 54656, 64654, 77501, 91368, 109003, 128244, 152279
Offset: 0

Views

Author

Alois P. Heinz, May 06 2012

Keywords

Examples

			a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(2) = 2: 2 = 1*1 + 1*1 = 1*2.
a(3) = 3: 3 = 1*1 + 1*1 + 1*1 = 1*1 + 1*2 = 1*3.
a(7) = 18 = A182269(7)-1, one of the 19 sums counted by A182269(7) is not allowed: 7 = 1*3 + 2*2.

Links

Crossrefs

Cf. A066739, A182269, A182270, A211856, A211857, A212215, A212216, A212217, A212218, A212219.

Programs

  • Maple
    with(numtheory):
b:= proc(n, m, i, j) option remember;
      `if`(n=0, 1, `if`(m<1, 0, b(n, m-1, i, j) +`if`(m>n, 0,
        add(b(n-m, m, min(i, k), min(j, m/k)), k=select(x->
         is(x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m<1, 0, b[n, m-1, i, j] + If[m>n, 0, Sum[b[n-m, m, Min[i, k], Min[j, m/k]], {k, Select[Divisors[m], # <= Min[Sqrt[m], i] && m <= j*# &]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 03 2014, after Alois P. Heinz *)

A212215 Number of representations of n as a sum of products of pairs of positive integers >=2, n = Sum_{k=1..m} i_k*j_k with 2<=i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_k*j_k<=i_{k+1}*j_{k+1}.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 5, 1, 4, 2, 9, 1, 11, 2, 16, 5, 18, 3, 33, 8, 31, 11, 52, 11, 64, 16, 83, 29, 100, 26, 152, 39, 159, 59, 233, 61, 280, 83, 354, 129, 423, 122, 591, 180, 644, 241, 864, 260, 1050, 341, 1282, 472, 1523, 490, 2016, 655, 2224
Offset: 0

Views

Author

Alois P. Heinz, May 06 2012

Keywords

Examples

			a(0) = 1: 0 = the empty sum.
a(4) = 1: 4 = 2*2.
a(6) = 1: 6 = 2*3.
a(8) = 2: 8 = 2*2 + 2*2 = 2*4.
a(9) = 1: 9 = 3*3.
a(10) = 2: 10 = 2*2 + 2*3 = 2*5.
a(17) = 1 = A182270(17)-1, one of the 2 sums counted by A182270(17) is not allowed: 17 = 2*4 + 3*3.

Links

Crossrefs

Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212216, A212217, A212218, A212219.

Programs

  • Maple
    with(numtheory):
b:= proc(n, m, i, j) option remember;
      `if`(n=0, 1, `if`(m<4, 0, b(n, m-1, i, j) +`if`(m>n, 0,
        add(b(n-m, m, min(i, k), min(j, m/k)), k=select(x->
         is(x>1 and x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m < 4, 0, b[n, m - 1, i, j] + If[m > n, 0, Sum[b[n - m, m, Min[i, k], Min[j, m/k]], {k, Select[ Divisors[m], # > 1 && # <= Min[Sqrt[m], i] && m <= j*# &]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 23 2017, after Alois P. Heinz *)

A212216 Number of representations of n as a sum of products of distinct pairs of positive integers, n = Sum_{k=1..m} i_k*j_k with i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_k*j_k

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 8, 12, 15, 18, 25, 28, 34, 44, 51, 59, 75, 87, 103, 124, 143, 163, 198, 228, 261, 310, 354, 404, 479, 538, 612, 708, 802, 907, 1049, 1175, 1320, 1518, 1711, 1910, 2187, 2431, 2724, 3097, 3447, 3843, 4348, 4818, 5373, 6032, 6693, 7420
Offset: 0

Views

Author

Alois P. Heinz, May 06 2012

Keywords

Examples

			a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(2) = 1: 2 = 1*2.
a(3) = 2: 3 = 1*1 + 1*2 = 1*3.
a(4) = 3: 4 = 1*1 + 1*3 = 1*4 = 2*2.
a(5) = 4: 5 = 1*2 + 1*3 = 1*1 + 1*4 = 1*1 + 2*2 = 1*5.
a(6) = 6: 6 = 1*1 + 1*2 + 1*3 = 1*2 + 1*4 = 1*2 + 2*2 = 1*1 + 1+5 = 1*6 = 2*3.

Links

Crossrefs

Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212217, A212218, A212219.

Programs

  • Maple
    with(numtheory):
b:= proc(n, m, i, j) option remember;
      `if`(n=0, 1, `if`(m<1, 0, b(n, m-1, i, j) +`if`(m>n, 0,
        add(b(n-m, m-1, min(i, k), min(j, m/k)), k=select(x->
         is(x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m<1, 0, b[n, m-1, i, j]+If[m>n, 0, Sum [b[n-m, m-1, Min[i, k], Min[j, m/k]], {k, Select[Divisors[m], # <= Min [Sqrt[m], i] && m <= j*# &]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 04 2014, after Alois P. Heinz *)

A212217 Number of representations of n as a sum of products of distinct pairs of positive integers >=2, n = Sum_{k=1..m} i_k*j_k with 2<=i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_k*j_k

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 3, 2, 5, 0, 7, 2, 8, 3, 10, 1, 15, 6, 14, 6, 21, 6, 28, 9, 26, 14, 38, 12, 50, 16, 47, 26, 70, 19, 82, 31, 87, 47, 111, 33, 141, 58, 143, 71, 182, 63, 228, 93, 231, 117, 289, 102, 364, 148, 354, 187, 462, 172, 537, 227
Offset: 0

Views

Author

Alois P. Heinz, May 06 2012

Keywords

Examples

			a(0) = 1: 0 = the empty sum.
a(4) = 1: 4 = 2*2.
a(6) = 1: 6 = 2*3.
a(8) = 1: 8 = 2*4.
a(9) = 1: 9 = 3*3.
a(10) = 2: 10 = 2*2 + 2*3 = 2*5.
a(12) = 3: 12 = 2*2 + 2*4 = 2*6 = 3*4.
a(13) = 1: 13 = 2*2 + 3*3.
a(14) = 3: 14 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.
a(15) = 2: 15 = 2*3 + 3*3 = 3*5.
a(16) = 5: 16 = 2*3 + 2*5 = 2*2 + 2*6 = 2*2 + 3*4 = 2*8 = 4*4.
a(19) = 2: 19 = 2*2 + 2*3 + 3*3 = 2*2 + 3*5.

Links

Crossrefs

Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212218, A212219.

Programs

  • Maple
    with(numtheory):
b:= proc(n, m, i, j) option remember;
      `if`(n=0, 1, `if`(m<4, 0, b(n, m-1, i, j) +`if`(m>n, 0,
        add(b(n-m, m-1, min(i, k), min(j, m/k)), k=select(x->
         is(x>1 and x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m < 4, 0, b[n, m - 1, i, j] + If[m > n, 0, Sum [b[n - m, m - 1, Min[i, k], Min[j, m/k]], {k, Select[Divisors[m], # > 1 && # <= Min [Sqrt[m], i] && m <= j*# &]}]]]];
a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 23 2017, after Alois P. Heinz *)

A212218 Number of representations of n as a sum of products of distinct pairs of positive integers, n = Sum_{k=1..m} i_k*j_k with i_k<=j_k, i_k

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 6, 5, 7, 7, 8, 9, 10, 9, 11, 12, 13, 14, 16, 14, 18, 21, 19, 20, 23, 23, 28, 28, 28, 30, 36, 33, 39, 42, 39, 44, 50, 46, 54, 57, 56, 62, 69, 64, 71, 77, 82, 85, 89, 84, 99, 107, 103, 111, 119, 117, 132, 137, 137, 142
Offset: 0

Views

Author

Alois P. Heinz, May 06 2012

Keywords

Examples

			a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(4) = 2: 4 = 1*4 = 2*2.
a(5) = 2: 5 = 1*1 + 2*2 = 1*5.
a(9) = 3: 9 = 1*1 + 2*4 = 1*9 = 3*3.
a(12) = 4: 12 = 1*2 + 2*5 = 1*12 = 2*6 = 3*4.
a(15) = 5: 15 = 1*3 + 2*6 = 1*3 + 3*4 = 1*1 + 2*7 = 1*15 = 3*5.

Links

Crossrefs

Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212219.

Programs

  • Maple
    with(numtheory):
b:= proc(n, m, i, j) option remember;
      `if`(n=0, 1, `if`(m<1, 0, b(n, m-1, i, j) +`if`(m>n, 0,
        add(b(n-m, m-1, min(i, k-1), min(j, m/k-1)), k=select(x->
         is(x<=min(sqrt(m), i) and m<=j*x), divisors(m))))))
    end:
a:= n-> b(n$4):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_, i_, j_] := b[n, m, i, j] = If[n == 0, 1, If[m<1, 0, b[n, m-1, i, j]+If[m>n, 0, Sum[b[n-m, m-1, Min[i, k-1], Min[j, m/k-1]], {k, Select[Divisors[m], # <= Min[Sqrt[m], i] && m <= j*#&]}]]]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 05 2014, after Alois P. Heinz *)

A282249 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k < j_k, j_k > j_{k+1} and all factors distinct.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 4, 4, 6, 5, 6, 8, 8, 9, 11, 10, 14, 15, 14, 14, 21, 18, 21, 25, 25, 30, 34, 33, 42, 45, 41, 55, 62, 58, 66, 79, 76, 94, 95, 97, 115, 131, 120, 148, 153, 159, 175, 203, 189, 226, 232, 243, 268, 299, 271, 340, 349, 363, 389
Offset: 0

Views

Author

Alois P. Heinz, Feb 09 2017

Keywords

Comments

Or number of partitions of n where part i has multiplicity < i and all multiplicities are distinct and different from all parts.

Examples

			a(0) = 1: the empty sum.
a(6) = 2: 1*6 = 2*3.
a(8) = 2: 1*8 = 2*4.
a(10) = 3: 1*10 = 2*5 = 1*4+2*3.
a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3.
a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4.
a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4.
a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2.
a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5.
a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.

Links

Crossrefs

Cf. A000009, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282379.

Programs

  • Maple
    h:= proc(n) option remember;
      (((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
    end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
                b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
     `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
      min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
  • Mathematica
    h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j, Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i - 1, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)

A282379 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k <= j_k, j_k > j_{k+1} and all factors distinct with the exception that i_k = j_k is allowed.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 5, 9, 9, 8, 11, 15, 13, 17, 17, 19, 24, 29, 23, 33, 37, 39, 40, 53, 48, 62, 63, 71, 77, 94, 81, 110, 116, 122, 123, 156, 152, 185, 180, 200, 213, 259, 236, 287, 298, 325, 333, 404, 386, 450, 457, 506, 531, 615, 579, 679, 721
Offset: 0

Views

Author

Alois P. Heinz, Feb 13 2017

Keywords

Examples

			a(4) = 2: 1*4 = 2*2.
a(5) = 2: 1*5 = 2*2+1*1.
a(6) = 2: 1*6 = 2*3.
a(7) = 3: 1*7 = 2*3+1*1 = 1*3+2*2.
a(8) = 3: 1*8 = 2*4 = 1*4+2*2.
a(9) = 4: 1*9 = 1*5+2*2 = 2*4+1*1 = 3*3.
a(10) = 5: 1*10 = 1*6+2*2 = 2*5 = 1*4+2*3 = 3*3+1*1.
a(11) = 6: 1*11 = 1*7+2*2 = 2*5+1*1 = 1*5+2*3 = 2*4+1*3 = 3*3+1*2.
a(12) = 5: 1*12 = 1*8+2*2 = 2*6 = 1*6+2*3 = 3*4.

Links

Crossrefs

Cf. A000009, A000330, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282249.

Programs

  • Maple
    h:= proc(n) option remember;
      n*(n+1)*(2*n+1)/6
    end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
                b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
     `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
      min(n-i*j, i-1), s union {j})), j=1..min(i, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
  • Mathematica
    h[n_] := h[n] = n(n+1)(2n+1)/6;
g[n_, i_, s_ ] := If[n == 0, 1, If[n > h[i], 0,
     b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] +
     If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j,
     Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)
Showing 1-7 of 7 results.

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