A062890 -id:A062890 - cp's OEIS frontend

A340247 Sum of the third largest parts r of the partitions of n into 4 parts q,r,s,t such that 1 <= q <= r <= s <= t and q + r + s > t.

0, 0, 0, 1, 1, 1, 3, 5, 7, 10, 15, 19, 28, 33, 46, 57, 75, 84, 113, 129, 164, 184, 232, 256, 319, 347, 425, 466, 561, 601, 723, 777, 918, 981, 1152, 1224, 1428, 1509, 1746, 1849, 2122, 2227, 2550, 2678, 3040, 3184, 3600, 3760, 4234, 4410, 4942, 5151, 5745, 5961, 6635

Offset: 1

Wesley Ivan Hurt, Jan 01 2021

nonn

Cf. A062890, A340246, A340248, A340249.

Table[Sum[Sum[Sum[j*Sign[Floor[(i + k + j)/(n - i - j - k + 1)]], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 60}]

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

A340248 Sum of the second largest parts s of the partitions of n into 4 parts q,r,s,t such that 1 <= q <= r <= s <= t and q + r + s > t.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 7, 10, 15, 22, 29, 42, 51, 71, 87, 116, 132, 177, 201, 259, 289, 368, 404, 508, 550, 681, 738, 900, 959, 1164, 1239, 1482, 1569, 1863, 1962, 2313, 2424, 2835, 2971, 3448, 3590, 4150, 4318, 4954, 5142, 5872, 6080, 6912, 7140, 8078, 8343, 9395, 9672

Offset: 1

Wesley Ivan Hurt, Jan 01 2021

nonn

Cf. A062890, A340246, A340247, A340249.

Table[Sum[Sum[Sum[i*Sign[Floor[(i + k + j)/(n - i - j - k + 1)]], {i, j,  Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 60}]

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

A340249 Sum of the largest parts t of the partitions of n into 4 parts q,r,s,t such that 1 <= q <= r <= s <= t and q + r + s > t.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 8, 14, 18, 30, 35, 56, 63, 95, 109, 156, 166, 235, 255, 346, 369, 491, 517, 676, 707, 907, 952, 1200, 1239, 1548, 1605, 1974, 2037, 2481, 2550, 3078, 3156, 3774, 3874, 4592, 4685, 5522, 5642, 6596, 6726, 7818, 7958, 9200, 9354, 10754, 10939, 12510

Offset: 1

Wesley Ivan Hurt, Jan 01 2021

nonn

Cf. A062890, A340246, A340247, A340248.

Table[Sum[Sum[Sum[(n - i - j - k) Sign[Floor[(i + k + j)/(n - i - j - k + 1)]], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 60}]

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

A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

Original entry on oeis.org

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

Offset: 0

Felix Huber, Jul 22 2025

nonn

a(n) is the number of partitions of n into 4 nonzero squares < n/2.

			The a(51) = 1 multiset is [1, 9, 16, 25].
The a(52) = 3 multisets are [1, 1, 25, 25], [4, 16, 16, 16] and [9, 9, 9, 25].

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A000290, A000414, A000534, A025428, A062890, A385736.

# After Alois P. Heinz (A025428)
b:=proc(n,i,t)
    option remember;
    `if`(n=0,`if`(t=0,1,0),`if`(i<1 or t<1, 0, b(n,i-1,t)+`if`(i^2>n,0,b(n-i^2,i,t-1))))
    end:
A385860:=n->b(n,floor(sqrt((n-1)/2)),4):
seq(A385860(n),n=0..87);

a(n) <= A025428(n).

A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 7, 7, 12, 19, 19, 25, 44, 72, 72, 119, 147, 152, 234, 292, 435, 777, 920, 946, 1135, 1161, 1377, 3702, 4293, 5942, 5942, 10741, 10741, 14483, 18953, 22091, 28658, 37686, 37686, 63053, 63053, 72389, 72389, 132732, 233773, 265312, 265312, 300443, 373266

Offset: 3

Felix Huber, Oct 13 2024

nonn

a(n) is the number of partitions of prime(n) into n prime parts < prime(n)/2.

First differs from A259254 at n=31: a(31) = 3702 but A259254(31) = 3703.

			a(7) = 2 because exactly the 2 partitions (2, 2, 2, 2, 3, 3, 3) and (2, 2, 2, 2, 2, 2, 5) have 7 prime parts and their sum is p(7) = 17.

Eric Weisstein's World of Mathematics, Polygon

Cf. A005044, A007042, A038499, A062890, A069906, A069907, A259254, A288253, A288254, A288255, A288256, A318604, A366398.

A376348:=proc(n)
   local a,p,x,i;
   a:=0;
   p:=ithprime(n);
   for x from NumberTheory:-pi(p/n)+1 to NumberTheory:-pi(p/2) do
      a:=a+numelems(select(i->nops(i)=n-1 and andmap(isprime,i),combinat:-partition(ithprime(n)-ithprime(x),ithprime(x))))
   od;
   return a
end proc;
seq(A376348(n),n=3..42);

a(n)={my(m=prime(n), p=primes(primepi((m-1)\2))); polcoef(polcoef(1/prod(i=1, #p, 1 - y*x^p[i], 1 + O(x*x^m)), m),n)} \\ Andrew Howroyd, Oct 13 2024

a(43) onwards from Andrew Howroyd, Oct 13 2024

cp's OEIS Frontend

A340247 Sum of the third largest parts r of the partitions of n into 4 parts q,r,s,t such that 1 <= q <= r <= s <= t and q + r + s > t.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A340248 Sum of the second largest parts s of the partitions of n into 4 parts q,r,s,t such that 1 <= q <= r <= s <= t and q + r + s > t.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A340249 Sum of the largest parts t of the partitions of n into 4 parts q,r,s,t such that 1 <= q <= r <= s <= t and q + r + s > t.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions