A369338 - OEIS

%I #4 Jan 20 2024 22:18:01

%S 0,0,2,3,9,11,20,24,34,46,54,64,77,101,107,129,135,175,170,217,221,

%T 271,252,325,303,386,372,454,405,546,464,594,569,676,611,803,665,851,

%U 803,968,819,1118,902,1180,1096,1261,1080,1482,1188,1522,1391,1669,1377,1878

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

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

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

%e 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.

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

%Y Cf. A369337.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Jan 20 2024