A350804 -id:A350804 - cp's OEIS frontend

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).

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.

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

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

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