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Number of pairs of divisors of n, (d1,d2), with d1 <= d2 such that d1 and d2 are nonconsecutive integers.
2

%I #20 Apr 15 2023 14:36:47

%S 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,

%T 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,

%U 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

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

%C 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.

%H Antti Karttunen, <a href="/A337362/b337362.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d1|n, d2|n, d1<=d2} (1 - [d1 + 1 = d2]), where [] is the Iverson bracket.

%F a(n) = A337363(n) + A000005(n).

%F a(n) = A184389(n) - A129308(n). - _Ridouane Oudra_, Apr 15 2023

%e 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.

%t 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}]

%o (PARI) a(n) = sumdiv(n, d1, sumdiv(n, d2, (d1<=d2) && (d1 + 1 != d2))); \\ _Michel Marcus_, Aug 25 2020

%Y Cf. A000005, A337363.

%Y Cf. also A335841, A337333, A184389, A129308.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Aug 24 2020