A026810 -id:A026810 - cp's OEIS frontend

A121895 Number of partitions of n into 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d.

0, 0, 0, 1, 1, 2, 2, 4, 2, 5, 4, 6, 4, 7, 5, 10, 5, 8, 6, 11, 8, 13, 6, 12, 7, 13, 9, 15, 8, 16, 10, 17, 10, 14, 10, 20, 11, 14, 10, 23, 10, 22, 12, 21, 15, 20, 8, 21, 12, 23, 18, 24, 11, 20, 15, 30, 18, 21, 8, 28, 14, 21, 18, 32, 16, 34, 16, 22, 15, 28, 14, 33, 14, 22, 20, 31, 18, 32, 15

Offset: 1

Zak Seidov, Sep 01 2006

nonn

			a(36)=20 because there are 20 partitions of 36 in 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d:
{33, 1, 1, 1}, {32, 2, 1, 1}, {30, 2, 2, 2}, {28, 4, 2, 2}, {27, 3, 3, 3}, {25, 5, 5, 1}, {24, 8, 2, 2}, {24, 6, 3, 3}, {24, 4, 4, 4}, {21, 7, 7, 1}, {20, 10, 5, 1}, {18, 6, 6, 6}, {17, 17, 1, 1}, {16, 16, 2, 2}, {16, 8, 8, 4}, {15, 15, 5, 1}, {15, 15, 3, 3}, {14, 14, 7, 1}, {12, 12, 6, 6}, {9, 9, 9, 9}.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A026810 = number of partitions of n into exactly 4 parts.
Column 4 of A122934.
Cf. A049822, A003238.

a(n) = Sum_{d|n, d>1} A122935(d-1). - Franklin T. Adams-Watters, Sep 20 2006

A321306 The number of connected weighted cubic graphs with weight n on 6 vertices.

Original entry on oeis.org

2, 2, 7, 12, 26, 41, 76, 113, 183, 264, 393, 543, 768, 1024, 1385, 1801, 2355, 2989, 3811, 4740, 5911, 7234, 8857, 10680, 12883, 15336, 18254, 21496, 25293, 29491, 34361, 39713, 45860, 52598, 60260, 68627, 78079, 88354, 99882, 112385, 126316, 141379, 158082, 176080

Offset: 6

R. J. Mathar, Nov 03 2018

Each vertex of the 2 simple cubic graphs is assigned an integer number (weight) >=1. The weight of the graph is the sum of the weights of the vertices.

The cycle indices of the permutation group of vertex permutations of the two cubic graphs on 6 vertices are ( +t[1]^6 +3*t[1]^2*t[2]^2 +2*t[3]^2 +4*t[2]^3 +2*t[6])/12 and +( +t[1]^6 +6*t[1]^4*t[2] +9*t[1]^2*t[2]^2 +4*t[1]^3*t[3] +12*t[1]*t[2]*t[3] +6*t[2]^3 +18*t[2]*t[4] +12*t[6] +4*t[3]^2)/72 . The ordinary generating function of the sequence is obtained by adding the two cycle indices and setting t[i] -> x^i/(1-x^i).

			a(6)=2 because there are 2 cubic graphs (see A002851), and if the weight is the same as the number of vertices, there is one case for each.

Andrew Howroyd, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,3,-2,1,1,-2,3,-2,0,-1,0,2,-1).

Cf. A026810 (4 vertices), A321307 (8 vertices), A005513.

G.f.: (x^10 +3*x^8 -x^7 +4*x^6 +4*x^4 +3*x^2 -2*x+2) *x^6/((-1+x)^6 *(1+x)^3 *(1+x^2) *(x^2+x+1)^2 *(x^2-x+1)).

Terms a(36) and beyond from Andrew Howroyd, Apr 27 2020

A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 13, 19, 14, 21, 19, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106

Offset: 3

Hugo Pfoertner, Apr 16 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A005044, A007237, A024153, A317182, A371070.

See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).

A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};

def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # David Radcliffe, Aug 01 2025

a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - Peter Munn, Jul 29 2025

b-file corrected by David Radcliffe, Aug 01 2025

A026924 Number of partitions of n into an odd number of parts, the greatest being 4; also, a(n+7) = number of partitions of n+3 into an even number of parts, each <=4.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 3, 3, 5, 5, 8, 8, 12, 13, 18, 19, 24, 26, 33, 35, 43, 46, 55, 59, 69, 74, 86, 91, 104, 111, 126, 134, 150, 159, 177, 187, 207, 219, 241, 254, 277, 292, 318, 334, 362, 380, 410, 430, 462, 484, 519, 542, 579, 605

Offset: 1

Clark Kimberling

nonn

R. J. Mathar, Table of n, a(n) for n = 1..394

4th column of A026920.

A026924 := proc(n)
    local a,p1,p2,p3,p4 ;
    a := 0 ;
    for p1 from 0 to n do
        for p2 from 0 to (n-p1)/2 do
            for p3 from op(1+modp(n-p1-2*p2,4),[0,3,2,1]) to (n-p1-2*p2)/3 by 4 do
                p4 := (n-p1-2*p2-3*p3)/4 ;
                if type(p4,'integer') and p4 >=1 and type(p1+p2+p3+p4,'odd') then
                    a := a+1 ;
                end if:
            end do:
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Aug 22 2019

a(n) + A026928(n) = A026810(n). - R. J. Mathar, Aug 22 2019

A340571 Number of partitions of n into 4 parts with at least one even part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 3, 6, 6, 11, 10, 18, 17, 27, 25, 39, 36, 54, 49, 72, 66, 94, 85, 120, 109, 150, 135, 185, 167, 225, 202, 270, 243, 321, 287, 378, 339, 441, 394, 511, 457, 588, 524, 672, 600, 764, 680, 864, 770, 972, 864, 1089, 969, 1215, 1079, 1350, 1200, 1495, 1326

Offset: 0

Wesley Ivan Hurt, Jan 11 2021

nonn

			a(5) = 1; [2,1,1,1];
a(7) = 3; [4,1,1,1], [3,2,1,1], [2,2,2,1];
a(9) = 6; [6,1,1,1], [5,2,1,1], [4,3,1,1], [4,2,2,1], [3,3,2,1], [3,2,2,2].

Index entries for sequences related to partitions

Cf. A026810, A340589.

Table[Sum[Sum[Sum[1 - Mod[k, 2] Mod[j, 2] Mod[i, 2] Mod[n - i - k - j, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (1 - (k mod 2) * (j mod 2) * (i mod 2) * ((n-i-j-k) mod 2)).

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign( ((k+1) mod 2) + ((j+1) mod 2) + ((i+1) mod 2) + ((n-i-j-k+1) mod 2) ).

A321307 The number of connected weighted cubic graphs with weight n on 8 vertices.

Original entry on oeis.org

5, 10, 41, 98, 257, 537, 1131, 2116, 3893, 6665, 11177, 17867, 28011, 42419, 63145, 91586, 130870, 183230, 253265, 344373, 463073, 614332, 807138, 1048517, 1350574, 1722948, 2181614, 2739523, 3417356, 4232137

Offset: 8

R. J. Mathar, Nov 03 2018

Each vertex of the 5 simple cubic graphs is assigned an integer number (weight) >=1. The weight of the graph is the sum of the weights of the vertices.

			a(8)=5 because there are 5 cubic graphs (see A002851), and if the weight is the same as the number of vertices, there is one case for each.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1,-2,5,-5,4,-3,-1,6,-6,4,-6,6,-1,-3,4,-5,5,-2,1,-1,-2,3,-1).

Cf. A026810 (4 vertices), A321306 (6 vertices).

G.f.: x^8*(x^18 +10*x^16 +5*x^15 +37*x^14 +8*x^13 +75*x^12 +16*x^11 +103*x^10 +16*x^9 +108*x^8 +13*x^7 +86*x^6 +3*x^5 +50*x^ 4+21*x^2 -5*x +5)/((-1+x)^8* (1+x)^4 *(x^2+x+1)^2 *(x^2-x+1) *(1+x^2)^2 *(1+x^4)).

A321379 Number of ways to write n as n = abc*d with 1 < a <= b <= c <= d < n.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Nov 08 2018

nonn

This sequence is different from A101638.

If p is prime, a(p^k) = A026810(k). - Robert Israel, Nov 08 2018

			16 = 2*2*2*2. So a(16) = 1.
24 = 2*2*2*3. So a(24) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A026810, A122179, A218320.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for a from 2 to floor(N^(1/4)) do
  for b from a to floor((N/a)^(1/3)) do
    for c from b to floor((N/a/b)^(1/2)) do
      for d from c to N/(a*b*c) do
        V[a*b*c*d]:= V[a*b*c*d]+1
od od od od:
convert(V,list); # Robert Israel, Nov 08 2018

A340572 Number of partitions of n into 4 parts with at least one prime part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 2, 5, 5, 8, 10, 13, 16, 21, 24, 31, 35, 41, 49, 57, 64, 75, 84, 95, 107, 119, 133, 147, 164, 179, 198, 215, 236, 256, 281, 300, 329, 349, 382, 407, 441, 465, 506, 531, 575, 603, 652, 681, 733, 765, 822, 853, 919, 952, 1019, 1057, 1128, 1166, 1247, 1284

Offset: 0

Wesley Ivan Hurt, Jan 11 2021

nonn

Index entries for sequences related to partitions

Cf. A010051, A026810, A340571.

b:= proc(n, i, t) option remember; series(
     `if`(n=0, t, `if`(i<1, 0, expand(x*b(n-i, min(n-i, i),
     `if`(isprime(i), 1, t)))+b(n, i-1, t))), x, 5)
    end:
a:= n-> coeff(b(n$2, 0), x, 4):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 24 2021

Table[Sum[Sum[Sum[Sign[(PrimePi[k] - PrimePi[k - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[i] - PrimePi[i - 1]) + (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1])], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign( c(k) + c(j) + c(i) + c(n-i-j-k) ), where c is the prime characteristic (A010051).

A340589 Number of partitions of n into 4 parts with at least one odd part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 11, 13, 18, 20, 27, 29, 39, 41, 54, 55, 72, 73, 94, 93, 120, 118, 150, 146, 185, 179, 225, 215, 270, 258, 321, 304, 378, 357, 441, 414, 511, 479, 588, 548, 672, 626, 764, 708, 864, 800, 972, 897, 1089, 1004, 1215, 1116, 1350, 1240, 1495

Offset: 0

Wesley Ivan Hurt, Jan 12 2021

nonn

			a(7) = 3; [4,1,1,1], [3,2,1,1], [2,2,2,1].
a(8) = 4; [5,1,1,1], [4,2,1,1], [3,3,1,1], [3,2,2,1], (not [2,2,2,2]).

Index entries for sequences related to partitions

Cf. A026810, A340571.

Table[Sum[Sum[Sum[Sign[Mod[k, 2] + Mod[j, 2] + Mod[i, 2] + Mod[n - i - j - k, 2]], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign((k mod 2) + (j mod 2) + (i mod 2) + ((n-i-j-k) mod 2)).

A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 4, 3, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 1, 5, 7, 2, 0, 1, 6, 8, 3, 0, 1, 6, 10, 5, 0, 1, 7, 12, 6, 1, 0, 1, 7, 14, 9, 1, 0, 1, 8, 16, 11, 2, 0, 1, 8, 19, 15, 3, 0, 1, 9, 21, 18, 5, 0, 1, 9, 24, 23, 7

Offset: 0

Seiichi Manyama, Apr 07 2025

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  1;
  0, 1, 3,  2;
  0, 1, 4,  3;
  0, 1, 4,  4, 1;
  0, 1, 5,  5, 1;
  0, 1, 5,  7, 2;
  0, 1, 6,  8, 3;
  0, 1, 6, 10, 5;
  0, 1, 7, 12, 6, 1;
  ...

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A000009.
Columns 0..10 give A000007, A000012, A004526(n-1), A069905(n-3), A026810(n-6), A026811(n-10), A026812(n-15), A026813(n-21), A026814(n-28), A026815(n-36), A026816(n-45).
Cf. A003056, A008284, A291954, A291960, A291968, A292047, A292049.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).

T(n,k) = |A292047(n,k)| = |A292049(n,k)|.

cp's OEIS Frontend

A121895 Number of partitions of n into 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d.

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A321306 The number of connected weighted cubic graphs with weight n on 6 vertices.

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A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

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PARI

Python

Formula

Extensions

A026924 Number of partitions of n into an odd number of parts, the greatest being 4; also, a(n+7) = number of partitions of n+3 into an even number of parts, each <=4.

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Maple

Formula

A340571 Number of partitions of n into 4 parts with at least one even part.

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Mathematica

Formula

A321307 The number of connected weighted cubic graphs with weight n on 8 vertices.

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Formula

A321379 Number of ways to write n as n = a*b*c*d with 1 < a <= b <= c <= d < n.

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Programs

Maple

A340572 Number of partitions of n into 4 parts with at least one prime part.

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Programs

Maple

Mathematica

Formula

A340589 Number of partitions of n into 4 parts with at least one odd part.

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A321379 Number of ways to write n as n = abc*d with 1 < a <= b <= c <= d < n.