A338021 -id:A338021 - cp's OEIS frontend

A337101 Number of partitions of n into two positive parts (s,t), s <= t, such that the harmonic mean of s and t is an integer.

0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 2, 2, 1, 1, 1, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 2, 3, 5, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 1, 4, 0, 1, 0, 1, 0, 1, 0, 6, 0, 1, 2, 1, 0, 1, 0, 2, 4, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 4, 0, 7

Offset: 1

Wesley Ivan Hurt, Aug 15 2020

Number of solutions, (s,t,k), to s^2 + t^2 = k*n such that s + t = n, 1 <= s <= t and 1 <= k <= n-1. - Wesley Ivan Hurt, Oct 01 2020

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

Cf. A004526, A337102, A337945, A338021, A338117.

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

A337101(n) = { my(u,t); sum(s=1, n\2, t = n-s; u = (s^2 + t^2); (!(u%n) && (u/n) <= n-1)); }; \\ Antti Karttunen, Dec 12 2021

a(n) = Sum_{i=1..floor(n/2)} (1 - ceiling(2*i*(n-i)/n) + floor(2*i*(n-i)/n)).
a(n) = A004526(n) - A337102(n).
a(n) = Sum_{i=1..floor(n/2)} Sum_{k=1..n-1} [i^2 + (n-i)^2 = n*k], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Oct 01 2020

A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116

Offset: 1

Wesley Ivan Hurt, Jan 16 2022

nonn

From Bernard Schott, Jan 22 2022: (Start)
A299174 is a subsequence because, if k = 2*u, we have s=t=u, s<=t, and u | u*k.
A082663 is another subsequence because, if k = p*q with p < q < 2p, then with s = k-p^2 = p*(q-p) and t = p^2, we have s <= t and p^2 | p*(q-p) * (pq).
It seems that A090196 is the subsequence of odd terms. (End)
gcd(s, t) > 1 where s and t and k > 2 are as in name. - David A. Corneth, Jan 22 2022
Numbers k such that k^2 has at least one divisor d with k/2 <= d < k. - Robert Israel, Jan 08 2025

			15 is in the sequence since 15 = 6+9 where 9 | 6*15 = 90.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A338021, A350804 (exactly one).
Subsequences: A082663, A299174.
Cf. A090196.

filter:= proc(n) nops(select(t -> t >= n/2 and t < n, numtheory:-divisors(n^2)))>=1 end proc:
select(filter, [$1..300]); # Robert Israel, Jan 08 2025

f(n) = sum(s=1, n\2, !((s*n)%(n-s))); \\ A338021
isok(k) = f(k) >= 1; \\ Michel Marcus, Jan 17 2022

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))).

A350804 Numbers k with exactly one partition into two parts (s,t), s<=t, such that t | s*k.

Original entry on oeis.org

2, 4, 8, 10, 14, 15, 16, 22, 26, 32, 34, 35, 38, 44, 46, 50, 52, 58, 62, 63, 64, 68, 74, 75, 76, 77, 82, 86, 91, 92, 94, 98, 99, 106, 116, 117, 118, 122, 124, 128, 134, 136, 142, 143, 146, 148, 152, 153, 158, 164, 166, 172, 175, 178, 184, 187, 188, 189, 194, 202, 206, 209

Offset: 1

Wesley Ivan Hurt, Jan 16 2022

nonn

Numbers k such that k^2 has exactly one divisor d with k/2 <= d < k. - Robert Israel, Jan 08 2025

			15 is in the sequence since 15 = 6+9 has exactly one partition into two parts (6,9) such that 9 | 6*15 = 90.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A338021, A350803 (at least one).

filter:= proc(n) nops(select(t -> t >= n/2 and t < n, numtheory:-divisors(n^2)))=1 end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 08 2025

f(n) = sum(s=1, n\2, !((s*n)%(n-s))); \\ A338021
isok(k) = f(k) == 1; \\ Michel Marcus, Jan 17 2022

A338528 Number of partitions of kn into two parts (s,t) such that s <= t and t | ks for k = 1..n.

Original entry on oeis.org

0, 2, 2, 5, 4, 11, 7, 13, 11, 19, 11, 27, 15, 28, 30, 34, 20, 45, 23, 52, 46, 49, 28, 71, 45, 58, 54, 78, 37, 105, 42, 81, 77, 79, 85, 124, 51, 90, 91, 137, 57, 156, 61, 134, 143, 115, 67, 178, 102, 160, 128, 162, 75, 187, 150, 206, 144, 143, 84, 276, 91, 156, 213, 199, 181, 263

Offset: 1

Wesley Ivan Hurt, Nov 07 2020

nonn

			a(6) = 11; The 11 partitions of 6*1, 6*2, ..., 6*6 into 2 parts (s,t) such that s <= t and t | k*s for k = 1..n are:
6: (3,3),
12: (4,8), (6,6),
18: (9,9),
24: (8,16), (12,12),
30: (5,25), (15,15),
36: (9,24), (12,24), (18,18).

Index entries for sequences related to partitions

Cf. A338021.

Table[Sum[Sum[(1 - Ceiling[k*i/(k*n - i)] + Floor[k*i/(k*n - i)]), {i, Floor[k*n/2]}], {k, n}], {n, 50}]

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

A350835 Area of the unique integer-sided rectangle with width W, length L, and semiperimeter S_n = L + W, such that L | W*S_n, where S_n = A350804(n).

Original entry on oeis.org

1, 4, 16, 25, 49, 54, 64, 121, 169, 256, 289, 250, 361, 484, 529, 625, 676, 841, 961, 686, 1024, 1156, 1369, 1350, 1444, 1372, 1681, 1849, 2058, 2116, 2209, 2401, 1458, 2809, 3364, 2916, 3481, 3721, 3844, 4096, 4489, 4624, 5041, 2662, 5329, 5476, 5776, 5832, 6241, 6724

Offset: 1

Wesley Ivan Hurt, Jan 17 2022

nonn

			a(6) = 54; 54 is the area of the unique 6 X 9 rectangle with semiperimeter A350804(6) = 15 = 6+9, such that 9 | 6*15 = 90.

Index entries for sequences related to partitions

Cf. A338021, A350803, A350804.

cp's OEIS Frontend

A337101 Number of partitions of n into two positive parts (s,t), s <= t, such that the harmonic mean of s and t is an integer.

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A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.

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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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A350804 Numbers k with exactly one partition into two parts (s,t), s<=t, such that t | s*k.

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A338528 Number of partitions of k*n into two parts (s,t) such that s <= t and t | k*s for k = 1..n.

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A350835 Area of the unique integer-sided rectangle with width W, length L, and semiperimeter S_n = L + W, such that L | W*S_n, where S_n = A350804(n).

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A338528 Number of partitions of kn into two parts (s,t) such that s <= t and t | ks for k = 1..n.