A369338 Sum of all the numbers less than each divisor d of n that do not divide d.

0, 0, 2, 3, 9, 11, 20, 24, 34, 46, 54, 64, 77, 101, 107, 129, 135, 175, 170, 217, 221, 271, 252, 325, 303, 386, 372, 454, 405, 546, 464, 594, 569, 676, 611, 803, 665, 851, 803, 968, 819, 1118, 902, 1180, 1096, 1261, 1080, 1482, 1188, 1522, 1391, 1669, 1377, 1878

Offset

1,3

Links

Table of n, a(n) for n=1..54.

Formula

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

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

Example

a(12) = 64. 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). The sum of the numbers is then: (2)+(3)+(4+5)+(5+7+8+9+10+11) = 64.

Mathematica

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

Crossrefs

Cf. A369337.

Sequence in context: A109658 A257027 A271548 * A110350 A057569 A177950

Adjacent sequences: A369335 A369336 A369337 * A369339 A369340 A369341

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 20 2024

Status

approved