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A337362
Number of pairs of divisors of n, (d1,d2), with d1 <= d2 such that d1 and d2 are nonconsecutive integers.
2
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
1,2
COMMENTS
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.
LINKS
FORMULA
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
EXAMPLE
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.
MATHEMATICA
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}]
PROG
(PARI) a(n) = sumdiv(n, d1, sumdiv(n, d2, (d1<=d2) && (d1 + 1 != d2))); \\ Michel Marcus, Aug 25 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Aug 24 2020
STATUS
approved