A034836 -id:A034836 - cp's OEIS frontend

A101427 Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.

1, 2, 8, 19, 42, 78, 139, 224, 350, 517, 744, 1032, 1405, 1862, 2432, 3115, 3942, 4914, 6067, 7400, 8954, 10729, 12768, 15072, 17689, 20618, 23912, 27571, 31650, 36150, 41131, 46592, 52598, 59149, 66312, 74088, 82549, 91694, 101600, 112267, 123774

Offset: 0

Anthony C Robin, Jan 17 2005

nonn

Subsequence of A034836, which gives the number of cuboids for volume n.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Cf. A034836, A101423, A101424, A101425, A101426.

Column k=3 of A277239.

a[n_] := Switch[Mod[n, 6], 0, n+1, 1|5, 3n/4 + 7/24, 2|4, n+2/3, 3, 3n/4 + 5/8] + n^4/24 + n^3/4 + 2n^2/3; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 06 2016, after Frederic Solbes' formula *)

a(n) = if (n % 3, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2)/24, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2+8)/24); \\ Michel Marcus, Mar 18 2014

If n is a multiple of 3, a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2+8)/24, otherwise a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2)/24. - Frederic Solbes, Mar 18 2014
G.f.: -(x^6+3*x^4+4*x^3+3*x^2+1)/((x^2+x+1)*(x+1)^2*(x-1)^5). - Colin Barker, Mar 27 2014
From Daniel Mondot, Sep 20 2016: (Start)
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) + 12, n>=8.
a(n) = 4*a(n-6) - 6*a(n-12) + 4*a(n-18) - a(n-24) + 1296, n>=24. (End)

Extended by Ray Chandler, Dec 17 2008
Edited by Ray Chandler, Dec 19 2008
a(0) = 1 prepended by Daniel Mondot, Sep 20 2016

A321359 Expansion of Product_{1 <= i <= j <= k} (1 + x^(ijk)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 16, 21, 27, 38, 49, 63, 84, 108, 137, 179, 226, 286, 365, 457, 570, 720, 894, 1106, 1378, 1700, 2087, 2577, 3151, 3847, 4707, 5723, 6941, 8439, 10197, 12300, 14852, 17863, 21433, 25740, 30797, 36794, 43963, 52372, 62288, 74098, 87905, 104149

Offset: 0

Seiichi Manyama, Nov 07 2018

nonn

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

Cf. A034836, A211856, A280473, A321360, A321361.

G.f.: Product_{k>0} (1 + x^k)^A034836(k).

A321360 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(ijk)).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 32, 44, 67, 91, 139, 186, 269, 362, 517, 686, 958, 1264, 1741, 2286, 3092, 4033, 5416, 7018, 9296, 11998, 15769, 20228, 26356, 33648, 43539, 55343, 71079, 89942, 114909, 144775, 183819, 230746, 291557, 364544, 458371, 571084, 714971, 887798, 1106704

Offset: 0

Seiichi Manyama, Nov 07 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A034836, A174465, A182269, A321359, A321361.

Euler transform of A034836.

G.f.: Product_{k>0} 1/(1 - x^k)^A034836(k).

A321361 Expansion of Product_{1 <= i <= j <= k} (1 - x^(ijk)).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 0, 2, -1, 0, 3, -1, -2, -1, 1, -6, 0, -1, -1, 0, 6, 1, 1, 0, 4, 0, 0, 10, -2, -1, -9, 7, -11, 13, -15, -7, -3, -9, 0, 6, -3, -9, 14, -9, 20, -17, 20, -2, 20, 1, 25, -9, 14, 13, -3, -7, -21, -9, -11, 6, -54, 39, -22, -30, -10, 35, -21, 8, -41, -23

Offset: 0

Seiichi Manyama, Nov 07 2018

sign

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

Convolution inverse of A321360.

Cf. A034836, A319359, A321299, A321359.

G.f.: Product_{k>0} (1 - x^k)^A034836(k).

A097203 Number of 4-tuples (a,b,c,d) with 1 <= a <= b <= c <= d, a^2+b^2+c^2+d^2 = n and gcd(a,b,c,d) = 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 2, 1, 1, 2, 1, 2, 0, 0, 3, 2, 1, 2, 1, 2, 0, 2, 2, 1, 3, 1, 2, 3, 0, 2, 4, 1, 2, 2, 1, 3, 0, 1, 3, 3, 2, 2, 4, 2, 0, 3, 2, 3, 3, 2, 3, 3, 0, 2, 5, 2, 3, 3, 2, 4, 0, 1, 5, 4, 2, 4, 2, 3, 0, 4, 4, 3

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Oct 22 2008

nonn

The old entry with this sequence number was a duplicate of A034836.
From Wolfdieter Lang, Mar 25 2013: (Start)
a(n) = 0 if n has no partition with four parts, each a (nonzero) square, and the parts have no common factor > 1.
n is not representable as a primitive sum of four nonzero squares.
If n' has a representation [s(1),s(2),s(3),s(4)] with 1 <= s(1) <= s(2) <= s(3) <= s(4) and sum(s(j)^2,j=1..4) = n', then [k*s(1),k*s(2),k*s(3),k*s(4)] is a representation of n := k^2*n'. Therefore, only primitive representations with gcd(s(1),s(2),s(3),s(4)) = 1 are here considered.
See A025428(n) for the multiplicity of the representations of n as a sum of four nonzero squares.
For the n values with a(n) not zero (primitively representable as a sum of four nonzero squares) see A222949. (End)

			The solutions (if any) for n <= 20 are as follows:
n = 1:
n = 2:
n = 3:
n = 4: 1 1 1 1
n = 5:
n = 6:
n = 7: 1 1 1 2
n = 8:
n = 9:
n = 10: 1 1 2 2
n = 11:
n = 12: 1 1 1 3
n = 13: 1 2 2 2
n = 14:
n = 15: 1 1 2 3
n = 16:
n = 17:
n = 18: 1 2 2 3
n = 19: 1 1 1 4
n = 20: 1 1 3 3
From _Wolfdieter Lang_, Mar 25 2013: (Start)
a(16) = 0 because 16 is not a primitive sum of four nonzero squares. The representation [2,2,2,2] of 16 is not primitive.
a(40) = 0 because the only representation as sum of four nonzero squares (A025428(4) = 1) is [2,2,4,4], but this is not primitive.
a(28) = 2 because the two primitive representations of 28 are
[1, 1, 1, 5] and [1, 3, 3, 3]. [2, 2, 2, 4] = 2*[1, 1, 1, 2] is not primitive due to 28 = 2^2*7. (End)

N. J. A. Sloane, Vinay Vaishampayan and Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 1..1024 from N. J. A. Sloane and Vinay Vaishampayan)

Cf. A025428, A000414, A222949.

b:= proc(n, i, g, t) option remember; `if`(n=0,
      `if`(g=1 and t=0, 1, 0), `if`(i<1 or t=0 or i^2*tn, 0, b(n-i^2, i, igcd(g, i), t-1))))
    end:
a:= n-> `if`(n<4, 0, b(n, isqrt(n-3), 0, 4)):
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 02 2013

Clear[b]; b[n_, i_, g_, t_] := b[n, i, g, t] = If[n == 0, If[g == 1 && t == 0, 1, 0], If[i < 1 || t == 0 || i^2*t < n, 0, b[n, i-1, g, t] + If[i^2 > n, 0, b[n-i^2, i, GCD[g, i], t-1]]]]; a[n_] := If[n < 4, 0, b[n, Sqrt[n-3] // Floor, 0, 4]]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Apr 05 2013, translated from Alois P. Heinz's Maple program *)

If a(n) > 0 then 8 does not divide n.

a(n) = k if there are k different quadruples [s(1),s(2),s(3),s(4)] with 1 <= s(1) <= s(2) <= s(3) <= s(4), gcd(s(1),s(2),s(3),s(4)) = 1 and sum(s(j)^2,j=1..4) = n. If there is no such quadruple then a(n) = 0. - Wolfdieter Lang, Mar 25 2013

A101423 Number of different cuboids with volume p^3 * q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

3, 6, 12, 19, 28, 38, 51, 64, 80, 97, 116, 136, 159, 182, 208, 235, 264, 294, 327, 360, 396, 433, 472, 512, 555, 598, 644, 691, 740, 790, 843, 896, 952, 1009, 1068, 1128, 1191, 1254, 1320, 1387, 1456, 1526, 1599, 1672, 1748, 1825, 1904, 1984, 2067, 2150, 2236

Offset: 0

Anthony C Robin, Jan 17 2005

nonn

Subsequence of A034836, which gives the number of cuboids for volume n.

Cf. A034836, A101424, A101425, A101426, A101427.

a(n) = A034836(2^3*3^n) = A034836(3^3*2^n) = A034836(p^3*q^n) for p,q distinct primes.

Empirical g.f.: -(x^3+3*x^2+3*x+3)/((x+1)*(x^2+x+1)*(x-1)^3). - Colin Barker, Mar 28 2014

Corrected, edited and extended by Ray Chandler, Dec 19 2008

a(0)=3 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

A101424 Number of different cuboids with volume p^4 * q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

4, 9, 18, 28, 42, 57, 76, 96, 120, 145, 174, 204, 238, 273, 312, 352, 396, 441, 490, 540, 594, 649, 708, 768, 832, 897, 966, 1036, 1110, 1185, 1264, 1344, 1428, 1513, 1602, 1692, 1786, 1881, 1980, 2080, 2184, 2289, 2398, 2508, 2622, 2737, 2856, 2976, 3100

Offset: 0

Anthony C Robin, Jan 17 2005

nonn

Subsequence of A034836, which gives the number of cuboids for volume n.

Cf. A034836, A101423, A101425, A101426, A101427.

a(n) = A034836(2^4*3^n) = A034836(3^4*2^n) = A034836(p^4*q^n) for p,q distinct primes.
From Colin Barker, Mar 28 2014: (Start)
The following is conjectured.
a(n) = (29 + 3*(-1)^n + 36*n + 10*n^2)/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -(x+4)/((x+1)*(x-1)^3). (End)

Edited and extended by Ray Chandler, Dec 19 2008

a(0)=4 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

A101425 Number of different cuboids with volume p^5 X q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

5, 12, 24, 38, 57, 78, 104, 132, 165, 200, 240, 282, 329, 378, 432, 488, 549, 612, 680, 750, 825, 902, 984, 1068, 1157, 1248, 1344, 1442, 1545, 1650, 1760, 1872, 1989, 2108, 2232, 2358, 2489, 2622, 2760, 2900, 3045, 3192, 3344, 3498, 3657, 3818, 3984, 4152

Offset: 0

Anthony C Robin, Jan 17 2005

nonn

Subsequence of A034836, which gives the number of cuboids for volume n.

Cf. A034836, A101423, A101424, A101426, A101427.

a(n) = A034836(2^5*3^n) = A034836(3^5*2^n) = A034836(p^5*q^n) for p,q distinct primes.
From Colin Barker, Mar 28 2014: (Start)
The following is conjectured.
a(n) = (37+3*(-1)^n+48*n+14*n^2)/8.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: -(2*x+5)/((x+1)*(x-1)^3). (End)

Edited and extended by Ray Chandler, Dec 19 2008

a(0)=5 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

A101426 Number of different cuboids with volume p^6 * q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

7, 16, 32, 51, 76, 104, 139, 176, 220, 267, 320, 376, 439, 504, 576, 651, 732, 816, 907, 1000, 1100, 1203, 1312, 1424, 1543, 1664, 1792, 1923, 2060, 2200, 2347, 2496, 2652, 2811, 2976, 3144, 3319, 3496, 3680, 3867, 4060, 4256, 4459, 4664, 4876, 5091, 5312

Offset: 0

Anthony C Robin, Jan 17 2005

nonn

Subsequence of A034836, which gives the number of cuboids for volume n.

Cf. A034836, A101423, A101424, A101425, A101427.

a(n) = A034836(2^6*3^n) = A034836(3^6*2^n) = A034836(p^6*q^n) for p,q distinct primes.

Empirical g.f.: -(3*x^3+9*x^2+9*x+7)/((x+1)*(x^2+x+1)*(x-1)^3). - Colin Barker, Mar 28 2014

Edited and extended by Ray Chandler, Dec 19 2008

a(0)=7 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

A226378 Number of distinct sums i+j+k with i,j,k >= 0, ijk = n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 11, 1, 2, 4, 4, 2, 5, 1, 9, 4, 2, 1, 10

Offset: 0

Robert Price, Jun 12 2013

nonn

			From _Antti Karttunen_, Aug 30 2017: (Start)
For n = 4 = 1*1*4 = 1*2*2, 1+1+4 = 6 and 1+2+2 = 5, so there are two distinct sums, and a(4) = 2.
For n = 6 = 1*1*6 = 1*2*3, 1+1+6 = 8 and 1+2+3 = 6, so there are two distinct sums, and a(6) = 2.
For n = 36, of its A034836(36) = 8 factorizations as x*y*z with 1 <= x <= y <= z: 1*1*36 = 1*2*18 = 1*3*12 = 1*4*9 = 1*6*6 = 2*2*9 = 2*3*6 = 3*3*4, sums 1+6+6 and 2+2+9 are both 13, while other triples yield unique sums, thus a(36) = 8-1 = 7. (End)

Robert Price and Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 100 terms from Robert Price).
Michael De Vlieger, Records and first positions of records in A226378(n), with 0 <= n <= 10^6.

Cf. A226357, A226359, A222945, A222947, A223133, A223134, A223135.
Cf. A008578 (gives the positions of 1's after a(0)=1).
Differs from A034836 for the first time at n=36.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]
(* Second program, more efficient: *)
{1}~Join~Table[With[{D = Divisors@ n}, Length@ Union@ Reap[Map[Function[a, Map[Function[b, Map[Function[c, If[a b c == n, Sow[a + b + c]]], Select[D, # <= n/a b &]]], Select[D, # <= n/a &]]], D]][[-1, 1]] ], {n, 100}] (* Michael De Vlieger, Aug 24 2017 *)

A226378(n) = { my(sums=Set()); if(!n,1,fordiv(n, i, for(j=i, (n/i), if(!(n%j),for(k=j, n/(i*j), if(i*j*k==n, sums = Set(concat(sums, (i+j+k)))))))); length(sums)); }; \\ Antti Karttunen, Aug 30 2017

For n >= 1, a(n) <= A034836(n). - Antti Karttunen, Aug 30 2017

cp's OEIS Frontend

A101427 Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.

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A321359 Expansion of Product_{1 <= i <= j <= k} (1 + x^(i*j*k)).

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A321360 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(i*j*k)).

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A321361 Expansion of Product_{1 <= i <= j <= k} (1 - x^(i*j*k)).

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A097203 Number of 4-tuples (a,b,c,d) with 1 <= a <= b <= c <= d, a^2+b^2+c^2+d^2 = n and gcd(a,b,c,d) = 1.

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A101423 Number of different cuboids with volume p^3 * q^n, where p,q are distinct prime numbers.

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A101424 Number of different cuboids with volume p^4 * q^n, where p,q are distinct prime numbers.

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A101425 Number of different cuboids with volume p^5 X q^n, where p,q are distinct prime numbers.

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A101426 Number of different cuboids with volume p^6 * q^n, where p,q are distinct prime numbers.

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A226378 Number of distinct sums i+j+k with i,j,k >= 0, i*j*k = n.

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A321359 Expansion of Product_{1 <= i <= j <= k} (1 + x^(ijk)).

A321360 Expansion of Product_{1 <= i <= j <= k} 1/(1 - x^(ijk)).

A321361 Expansion of Product_{1 <= i <= j <= k} (1 - x^(ijk)).

A226378 Number of distinct sums i+j+k with i,j,k >= 0, ijk = n.