A335841 -id:A335841 - cp's OEIS frontend

A337362 Number of pairs of divisors of n, (d1,d2), with d1 <= d2 such that d1 and d2 are nonconsecutive integers.

1, 2, 3, 5, 3, 8, 3, 9, 6, 9, 3, 18, 3, 9, 10, 14, 3, 19, 3, 19, 10, 9, 3, 33, 6, 9, 10, 20, 3, 33, 3, 20, 10, 9, 10, 42, 3, 9, 10, 34, 3, 33, 3, 20, 21, 9, 3, 52, 6, 20, 10, 20, 3, 34, 10, 34, 10, 9, 3, 73, 3, 9, 21, 27, 10, 34, 3, 20, 10, 35, 3, 74, 3, 9, 21, 20, 10, 34, 3, 53, 15

Offset: 1

Wesley Ivan Hurt, Aug 24 2020

nonn

Number of distinct rectangles that can be made using the divisors of n as side lengths and whose length is never one more than its width.

			a(6) = 8; The divisors of 6 are {1,2,3,6}. There are 8 divisor pairs, (d1,d2), with d1 <= d2 that do not contain consecutive integers. They are (1,1), (1,3), (1,6), (2,2), (2,6), (3,3), (3,6) and (6,6). So a(6) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A337363.

Cf. also A335841, A337333, A184389, A129308.

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

a(n) = sumdiv(n, d1, sumdiv(n, d2, (d1<=d2) && (d1 + 1 != d2))); \\ Michel Marcus, Aug 25 2020

a(n) = Sum_{d1|n, d2|n, d1<=d2} (1 - [d1 + 1 = d2]), where [] is the Iverson bracket.
a(n) = A337363(n) + A000005(n).
a(n) = A184389(n) - A129308(n). - Ridouane Oudra, Apr 15 2023

A337333 Number of pairs of odd divisors of n, (d1,d2), such that d1 <= d2.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 3, 1, 6, 3, 3, 3, 3, 3, 10, 1, 3, 6, 3, 3, 10, 3, 3, 3, 6, 3, 10, 3, 3, 10, 3, 1, 10, 3, 10, 6, 3, 3, 10, 3, 3, 10, 3, 3, 21, 3, 3, 3, 6, 6, 10, 3, 3, 10, 10, 3, 10, 3, 3, 10, 3, 3, 21, 1, 10, 10, 3, 3, 10, 10, 3, 6, 3, 3, 21, 3, 10, 10, 3, 3, 15, 3, 3, 10, 10

Offset: 1

Wesley Ivan Hurt, Aug 23 2020

nonn

Number of distinct rectangles that can be made whose side lengths are odd divisors of n.

			a(15) = 10; There are 10 pairs of odd divisors of 15, (d1,d2), such that d1<=d2. They are: (1,1), (1,3), (1,5), (1,15), (3,3), (3,5), (3,15), (5,5), (5,15), (15,15). So a(15) = 10.
a(16) = 1; (1,1) is the only pair of odd divisors of 16, (d1,d2), such that d1<=d2. So a(16) = 1.
a(17) = 3; There are 3 pairs of odd divisors of 17, (d1,d2), such that d1<=d2. They are (1,1), (1,17) and (17,17). So a(17) = 3.
a(18) = 6; There are 6 pairs of odd divisors of 18, (d1,d2), such that d1<=d2. They are: (1,1), (1,3), (1,9), (3,3), (3,9) and (9,9). So a(18) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000079, A000217, A001227 (number of odd divisors), A335841.

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

A337333(n) = binomial(numdiv(n>>valuation(n,2))+1,2); \\ Antti Karttunen, Dec 12 2021

a(n) = Sum_{d1|n, d2|n, d1 and d2 odd, d1<=d2} 1.
From Bernard Schott, Aug 24 2020: (Start)
a(n) = 1 if and only if n = 2^k, k >= 0 (A000079).
a(n) = 3 if n is an odd prime. (End)
a(n) = A000217(A001227(n)). - Antti Karttunen, Dec 12 2021

cp's OEIS Frontend

A337362 Number of pairs of divisors of n, (d1,d2), with d1 <= d2 such that d1 and d2 are nonconsecutive integers.

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