A369337 Total number of numbers less than each divisor d of n that do not divide d.

0, 0, 1, 1, 3, 3, 5, 5, 7, 9, 9, 10, 11, 15, 15, 16, 15, 21, 17, 24, 23, 27, 21, 30, 25, 33, 30, 38, 27, 45, 29, 42, 39, 45, 39, 55, 35, 51, 47, 60, 39, 69, 41, 66, 60, 63, 45, 79, 51, 75, 63, 80, 51, 90, 63, 90, 71, 81, 57, 114, 59, 87, 86, 99, 75, 117, 65, 108, 87, 117, 69, 135, 71, 105

Offset

1,5

Links

Table of n, a(n) for n=1..74.

Formula

a(n) = Sum_{d|n} Sum_{k=1..d} ceiling(d/k) - floor(d/k).

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

Example

a(12) = 10. The divisors of 12 are {1,2,3,4,6,12}. There are 0 numbers (positive integers) less than 1, 0 numbers less than 2 not dividing 2, 1 number less than 3 not dividing 3 (namely 2), 1 number less than 4 not dividing 4 (namely 3), 2 numbers less than 6 not dividing 6 (namely 4 and 5) and 6 numbers less than 12 not dividing 12 (namely 5,7,8,9,10,11).

Mathematica

Table[Sum[Sum[(Ceiling[d/k] - Floor[d/k]), {k, d}], {d, Divisors[n]}], {n,
  100}]

Crossrefs

Cf. A369338.

Sequence in context: A339102 A095878 A341143 * A157966 A212597 A268188

Adjacent sequences: A369334 A369335 A369336 * A369338 A369339 A369340

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 20 2024

Status

approved