A337101 -id:A337101 - cp's OEIS frontend

A338021 Number of partitions of n into two parts (s,t) such that s <= t and t | s*n.

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 1, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 1, 3, 0, 1, 0, 3, 0, 3, 0, 1, 2, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 6, 0, 1, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 1, 1, 1, 2, 0, 3, 0, 1, 0, 6, 0, 1, 0, 2, 0, 6, 1, 1, 0, 1, 0, 3, 0, 1, 1, 2, 0, 2, 0, 2, 2

Offset: 1

Wesley Ivan Hurt, Oct 06 2020

nonn

			a(6) = 2; The partitions of 6 into 2 parts are (1,5), (2,4) and (3,3). Since 4 | 2*6 = 12 and 3 | 3*6 = 18, we have two such partitions.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to partitions

Cf. also A337101, A338117.

Table[Sum[(1 - Ceiling[n*i/(n - i)] + Floor[n*i/(n - i)]), {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}],?(#[[2]]<=#[[1]]&&Mod[n #[[2]],#[[1]]]==0&)],{n,110}] (* _Harvey P. Dale, Jun 14 2025 *)

A338021(n) = sum(s=1,n\2,!((s*n)%(n-s))); \\ Antti Karttunen, Dec 12 2021

a(n) = Sum_{i=1..floor(n/2)} (1 - ceiling(n*i/(n-i)) + floor(n*i/(n-i))).

Data section extended up to 105 terms by Antti Karttunen, Dec 12 2021

A337945 Numbers m with a solution (s,t,k) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 118, 120

Offset: 1

Wesley Ivan Hurt, Oct 01 2020

nonn

			8 is in the sequence since it has the solutions (s,t,k) = (4,4,4) and (2,6,5) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.
9 is in the sequence since it has the solution (s,t,k) = (3,6,5) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.

Cf. A337101, A160014, A039956.

# Quite inefficient compared to the conjectured formula.
KD := (n, k) -> Physics:-KroneckerDelta[n, k]:
S := k -> local i, j; add(add(KD((i^2 + (k - i)^2)/j , k), j = 1..k-1),
i = 1..floor(k/2)): select(k -> S(k) > 0, [seq(k, k = 1..40)]); # Peter Luschny, Jun 08 2023

Table[If[Sum[Sum[KroneckerDelta[(i^2 + (n - i)^2)/k, n], {k, n - 1}], {i, Floor[n/2]}] > 0, n, {}], {n, 120}] // Flatten

k is a term <=> Sum_{i=1..floor(k/2)} Sum_{j=1..k-1} KroneckerDelta((i^2 + (k - i)^2)/j, k) > 0.

Conjecture: k is a term <=> k * Clausen(k, 1) <> 2 * Clausen(k, 0), (Clausen = A160014). In other words: k is in this sequence iff it is not an odd squarefree number. - Peter Luschny, Jun 08 2023

A338117 Number of partitions of n into two parts (s,t) such that (t-s) | n, where s < t.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 3, 1, 2, 1, 3, 3, 1, 1, 5, 2, 1, 3, 3, 1, 3, 1, 4, 3, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 3, 3, 1, 3, 3, 5, 3, 1, 1, 7, 1, 1, 5, 5, 3, 3, 1, 3, 3, 3, 1, 8, 1, 1, 5, 3, 3, 3, 1, 7, 4, 1, 1, 7, 3, 1, 3, 5, 1, 5, 3, 3, 3

Offset: 1

Wesley Ivan Hurt, Oct 10 2020

Apparently a(n) = A320111(n) - 1. - Hugo Pfoertner, Oct 30 2020

The above observation is true, which can be seen from the formula A320111(2n) = A000005(n), A320111(2n+1) = A000005(2n+1). For odd numbers, the difference (t-s) may range over all the divisors of n except the n itself, and for even numbers the difference (t-s) [which is always even] may range only over the even divisors of n, except the n itself. Note that A000005(2n) = A000005(n) + A001227(n). - Antti Karttunen, Dec 12 2021

			a(8) = 2; The partitions of 8 into two parts (s,t) such that s < t are (7,1), (6,2), (5,3) and (4,4). Only the partitions (6,2) and (5,3) have (6-2) | 8 and (5-3) | 8, so a(8) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..20160
Index entries for sequences related to partitions

Cf. A000005, A001227, A320111.

Cf. also A337101, A338021.

Table[Sum[(1 - Ceiling[n/(n - 2 i)] + Floor[n/(n - 2 i)]), {i, Floor[(n - 1)/2]}], {n, 100}]

for(n=1,85,my(j=0);forpart(x=n,if(#x==2,if(x[2]!=x[1]&&!(n%(x[2]-x[1])),j++)));print1(j,", ")) \\ Hugo Pfoertner, Oct 30 2020

A338117(n) = sum(s=1,(n-1)\2,!(n%(n-(2*s)))); \\ Antti Karttunen, Dec 12 2021

a(n) = Sum_{i=1..floor((n-1)/2)} (1 - ceiling(n/(n-2*i)) + floor(n/(n-2*i))).

A337102 Number of partitions of n into two positive integer parts (s,t), s<=t, such that the harmonic mean of the smallest and largest part is not an integer.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 2, 3, 4, 5, 5, 6, 6, 7, 6, 8, 6, 9, 9, 10, 10, 11, 10, 10, 12, 12, 13, 14, 14, 15, 12, 16, 16, 17, 15, 18, 18, 19, 18, 20, 20, 21, 21, 21, 22, 23, 22, 21, 20, 25, 25, 26, 24, 27, 26, 28, 28, 29, 29, 30, 30, 30, 28, 32, 32, 33, 33, 34, 34, 35, 30, 36, 36, 35, 37, 38, 38

Offset: 1

Wesley Ivan Hurt, Aug 15 2020

Index entries for sequences related to partitions

Cf. A004526, A337101.

Table[Sum[Ceiling[2*i*(n - i)/n] - Floor[2*i*(n - i)/n], {i, Floor[n/2]}], {n, 100}]

a(n) = Sum_{i=1..floor(n/2)} (ceiling(2*i*(n-i)/n) - floor(2*i*(n-i)/n)).

a(n) = A004526(n) - A337101(n).

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A338021 Number of partitions of n into two parts (s,t) such that s <= t and t | s*n.

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A337945 Numbers m with a solution (s,t,k) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.

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A338117 Number of partitions of n into two parts (s,t) such that (t-s) | n, where s < t.

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A337102 Number of partitions of n into two positive integer parts (s,t), s<=t, such that the harmonic mean of the smallest and largest part is not an integer.

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