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 1-10 of 19 results. Next

A211856 Number of representations of n as a sum of products of distinct pairs of positive integers, considered to be equivalent when terms or factors are reordered.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 10, 15, 20, 25, 34, 44, 56, 74, 94, 117, 151, 190, 236, 298, 370, 455, 567, 699, 853, 1050, 1282, 1555, 1898, 2299, 2770, 3351, 4035, 4837, 5811, 6952, 8288, 9898, 11782, 13978, 16600, 19660, 23225, 27451, 32366, 38074, 44799, 52609
Offset: 0

Views

Author

Alois P. Heinz, Apr 22 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 = 2*2 = 1*1 + 1*3 = 1*4.
a(5) = 4: 5 = 1*1 + 2*2 = 1*2 + 1*3 = 1*1 + 1*4 = 1*5.
a(6) = 6: 6 = 1*1 + 1*5 = 1*1 + 1*2 + 1*3 = 1*2 + 1*4 = 1*2 + 2*2 = 1*6 = 2*3
a(7) = 8: 7 = 1*1 + 1*2 + 1*4 = 1*1 + 1*2 + 2*2 = 1*1 + 1*6 = 1*1 + 2*3 = 1*2 + 1*5 = 1*3 + 1*4 = 1*3 + 2*2 = 1*7.

Links

Crossrefs

Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211857.

Programs

  • Maple
    with(numtheory):
b:= proc(n, i) option remember; local c;
      c:= ceil(tau(i)/2);
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
       +add(b(n-i*j, i-1) *binomial(c, j), j=1..min(c, n/i))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..60);
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{c}, c = Ceiling[DivisorSigma[0, i]/2]; If[n == 0, 1, If[i < 1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] *Binomial[c, j], {j, 1, Min[c, n/i]}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[Product[(1 + x^(k*j)), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2019 *)

Formula

G.f.: Product_{k>0} (1+x^k)^A038548(k). - Vaclav Kotesovec, Aug 19 2019
G.f.: Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))). - Vaclav Kotesovec, Aug 19 2019

A182270 Number of representations of n as a sum of products of pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 5, 1, 4, 2, 9, 2, 11, 3, 16, 7, 19, 6, 34, 13, 35, 18, 57, 23, 73, 32, 99, 53, 125, 60, 186, 92, 215, 127, 311, 164, 394, 221, 518, 320, 656, 386, 903, 545, 1091, 719, 1470, 925, 1863, 1215, 2390, 1642, 3015, 2037, 3966
Offset: 0

Views

Author

Alois P. Heinz, Apr 22 2012

Keywords

Examples

			a(0) = 1: 0 = the empty sum.
a(1) = a(2) = a(3) = 0: no product is < 4.
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(12) = 5: 12 = 2*2 + 2*2 + 2*2 = 2*2 + 2*4 = 2*3 + 2*3 = 2*6 = 3*4.
a(13) = 1: 13 = 2*2 + 3*3.
a(14) = 4: 14 = 2*2 + 2*2 + 2*3 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.

Links

Crossrefs

Cf. A000005, A006171, A038548, A066739, A182269, A211856, A211857.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
       d*(ceil(tau(d)/2)-1), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*(Ceiling[DivisorSigma[0, d]/2] - 1), {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)

Formula

Euler transform of A038548-1.
G.f.: Product_{k>0} 1/(1-x^k)^(A038548(k)-1).
G.f.: Product_{i>=1} Product_{j=2..i} 1/(1 - x^(i*j)). - Ilya Gutkovskiy, Sep 23 2019

A211857 Number of representations of n as a sum of products of distinct pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 3, 2, 5, 1, 7, 3, 8, 5, 11, 4, 16, 9, 17, 12, 25, 13, 34, 20, 37, 28, 53, 32, 69, 46, 78, 63, 108, 71, 136, 100, 160, 134, 210, 152, 265, 211, 313, 268, 403, 316, 506, 421, 596, 528, 759, 629, 943, 814, 1111, 1016
Offset: 0

Views

Author

Alois P. Heinz, Apr 22 2012

Keywords

Examples

			a(0) = 1: 0 = the empty sum.
a(1) = a(2) = a(3) = 0: no product is < 4.
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(16) = 5: 16 = 2*2 + 2*6 = 2*2 + 3*4 = 2*3 + 2*5 = 2*8 = 4*4.

Links

Crossrefs

Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211856.

Programs

  • Maple
    with(numtheory):
b:= proc(n, i) option remember; local c;
      c:= ceil(tau(i)/2)-1;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)
       +add(b(n-i*j, i-1) *binomial(c, j), j=1..min(c, n/i))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{c}, c = Ceiling[DivisorSigma[0, i]/2]-1; If[n==0, 1, If[i<2, 0, b[n, i-1]+Sum[b[n-i*j, i-1]*Binomial[c, j], {j, 1, Min[c, n/i]}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

Formula

G.f.: Product_{k>0} (1+x^k)^(A038548(k)-1). - Vaclav Kotesovec, Aug 19 2019
G.f.: Product_{i>=1} Product_{j=2..i} (1 + x^(i*j)). - Ilya Gutkovskiy, Sep 23 2019

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 *)

A212219 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

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 3, 0, 3, 1, 3, 2, 3, 1, 5, 3, 3, 2, 6, 4, 5, 3, 6, 6, 7, 2, 11, 5, 8, 6, 12, 7, 10, 8, 12, 11, 14, 8, 17, 11, 16, 13, 19, 13, 23, 15, 22, 17, 25, 18, 29, 24, 24, 23, 36, 25, 34, 25, 42, 34, 39, 30, 47, 40, 48, 37
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(12) = 2: 12 = 2*6 = 3*4.
a(13) = 1: 13 = 2*2 + 3*3.
a(20) = 3: 20 = 2*2 + 4*4 = 2*10 = 4*5.
a(23) = 1: 23 = 2*4 + 3*5.
a(31) = 3: 31 = 2*5 + 3*7 = 2*3 + 5*5 = 2*2 + 3*9.

Links

Crossrefs

Cf. A066739, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218.

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-1), min(j, m/k-1)), 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-1], Min[j, m/k-1]], {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, Dec 09 2014, after Alois P. Heinz *)

A321299 Expansion of Product_{1 <= i <= j} (1 - x^(i*j)).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 0, 2, 0, -1, 2, -1, -2, 0, 0, -4, 0, -1, 1, 0, 2, 2, 2, -1, 3, -1, 1, 6, -2, 1, -6, 1, -4, 5, -7, -5, -1, -4, -2, 6, -4, 0, 0, -2, 13, -7, 8, 2, 5, 1, 10, 2, 9, -3, -1, -1, -6, -7, 2, -5, -8, 1, -6, -17, -4, 13, -8, 1, -16, -14, 3, 7, -1, 20, -10, 13, -3, 8
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2018

Keywords

Links

Crossrefs

Convolution inverse of A182269.
Cf. A038548, A211856, A321300.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 - x^k)^Floor[(DivisorSigma[0, k] + 1)/2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 03 2018 *)

Formula

G.f.: Product_{k>0} (1 - x^k)^A038548(k).
Showing 1-10 of 19 results. Next

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