A211856 -id:A211856 - cp's OEIS frontend

A182269 Number of representations of n as a sum of products of pairs of positive integers, considered to be equivalent when terms or factors are reordered.

1, 1, 2, 3, 6, 8, 14, 19, 31, 43, 65, 88, 132, 177, 253, 340, 478, 633, 874, 1150, 1562, 2045, 2736, 3553, 4713, 6082, 7969, 10234, 13301, 16973, 21889, 27789, 35577, 44961, 57179, 71906, 90950, 113874, 143204, 178592, 223505, 277599, 345822, 427934, 530797

Offset: 0

Alois P. Heinz, Apr 22 2012

nonn

From Yifan Xie, Sep 25 2024: (Start)
a(n) is the number of distinct sets A = {b_1, b_2, ..., b_n} such that 2*n positive integers x_1, x_2, ..., x_(2*n) exist where A = {x_1+x_2, x_3+x_4, ..., x_(2*n-1)+x_(2*n)} = {x_1*x_2, x_3*x_4, ..., x_(2*n-1)*x_(2*n)}.
Proof: Suppose that the number of sets A is b(n). Denote (x_(2*i-1)-1)*(x_(2*i)-1) by c_i. (1 <= i <= n)
Taking the sums of B and C, c_1 + c_2 + ... + c_n = n. (1)
Consider b_1, ..., b_n as n vertices, then the map B -> C is a directed graph G on these vertices, where each vertex has a source and a sink, so it can either be a cycle itself or decomposed into two or more cycles.
For the first case, the condition is equivalent to every proper subset of A' = {b_1, ..., b_n} is invalid for the corresponding n. Using (1), every partial sum of c_i is not equal to the number of addends. Therefore, c_i != 1. Then there must exist c_j = 0, hence c_i != 2. Then there must exist another c_k = 0, hence c_i != 3, and so on. Thus c_1, c_2, ..., c_n must be a permutation of 0, 0, ..., 0, n. Suppose that c_n = n, x_1 = x_3 = ... = x_(2*n-3) = 1. Since n has floor((A000005(n)+1)/2) ways to be expressed as the product of two positive integers, each product n = y*z means that x_(2*i-1) = y+1, x_(2*i) = z+1, thus there exists (y+1)*(z+1) in A, 1 + x_(2*l) = (a+1)*(b+1), 1*x_(2*l) = (a+1)*(b+1)-1, there exists (a+1)*(b+1)-1 in A, and so on until a+b+2 = (a+1)*(b+1)-1. In conclusion, there are floor((A000005(n)+1)/2) distinct A's in the second case, each of which is a group of consecutive integers. Denote the array by n = a*b .
For the second case, the array A can be decomposed into smaller arrays representing smaller n's, without breaking the structures of B and C. This process will finally end with all smaller arrays in the first case. Using the same notation, the arrays can be expressed as n = y_1*z_1 + y_2*z_2 + ... + y_s*z*s.
Therefore b(n) is the number of representations of n as a sum of products of pairs of unordered positive integers, hence b(n) = a(n). (End)

			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(4) = 6: 4 = 1*1 + 1*1 + 1*1 + 1*1 = 1*1 + 1*1 + 1*2 = 1*1 + 1*3 = 1*2 + 1*2 = 2*2 = 1*4.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A000005, A006171, A038548, A066739, A182270, A211856, A211857.

with(numtheory):
a:= proc(n) option remember;  `if`(n=0, 1, add(add(
       d*ceil(tau(d)/2), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Ceiling[DivisorSigma[0, d]/2], {d, Divisors[j]}]*a[n-j], {j, 1, 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/(1 - x^(k*j)), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2019 *)

Euler transform of A038548.
G.f.: Product_{k>0} 1/(1-x^k)^A038548(k).
G.f.: Product_{k>=1} (Product_{j=1..k} 1/(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

Alois P. Heinz, Apr 22 2012

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

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

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);

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 *)

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

Alois P. Heinz, Apr 22 2012

nonn

			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.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

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

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);

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 *)

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_kj_k with i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_kj_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

Alois P. Heinz, May 06 2012

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

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);

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_kj_k with 2<=i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_kj_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

Alois P. Heinz, May 06 2012

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

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);

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_kj_k with i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_kj_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

Alois P. Heinz, May 06 2012

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

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);

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_kj_k with 2<=i_k<=j_k, i_k<=i_{k+1}, j_k<=j_{k+1}, i_kj_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

Alois P. Heinz, May 06 2012

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

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);

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

Alois P. Heinz, May 06 2012

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

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);

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

Alois P. Heinz, May 06 2012

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

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);

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

Seiichi Manyama, Nov 03 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A182269.

Cf. A038548, A211856, A321300.

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 *)

G.f.: Product_{k>0} (1 - x^k)^A038548(k).

cp's OEIS Frontend

A182269 Number of representations of n as a sum of products of pairs of positive integers, considered to be equivalent when terms or factors are reordered.

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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.

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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.

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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}.

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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}.

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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

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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

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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

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

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

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

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