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A355750
Sum of the divisors of 2n minus the number of divisors of 2n.
1
1, 4, 8, 11, 14, 22, 20, 26, 33, 36, 32, 52, 38, 50, 64, 57, 50, 82, 56, 82, 88, 78, 68, 114, 87, 92, 112, 112, 86, 156, 92, 120, 136, 120, 136, 183, 110, 134, 160, 176, 122, 212, 128, 172, 222, 162, 140, 240, 165, 208, 208, 202, 158, 268, 208, 238, 232, 204, 176, 344, 182
OFFSET
1,2
COMMENTS
Consider the partitions of 2n into 2 parts (s,t), where s <= t. a(n) gives the sum of all the quotients t/s such that t/s is an integer. (See example.)
FORMULA
a(n) = sigma(2n) - tau(2n).
a(n) = Sum_{d|2n} (2n-d)/d.
a(n) = A065608(2n) = A000203(2n) - A000005(2n).
a(n) = A062731(n) - A099777(n).
a(n) = Sum_{k=1..n} m*c(m), where m=(2n-k)/k and c(m)=1-ceiling(m)+floor(m).
EXAMPLE
a(7) = 20; the partitions of 2*7 = 14 into two parts (s,t) where s <= t are: (1,13), (2,12), (3,11), (4,10), (5,9), (6,8), and (7,7). The sum of the quotients t/s such that each t/s is an integer is then: 13/1 + 12/2 + 7/7 = 13 + 6 + 1 = 20.
MATHEMATICA
Table[DivisorSigma[1, 2 n] - DivisorSigma[0, 2 n], {n, 80}]
PROG
(PARI) a(n) = my(f=factor(2*n)); sigma(f) - numdiv(f); \\ Michel Marcus, Jul 16 2022
CROSSREFS
Cf. A000005 (tau), A000203 (sigma), A062731, A099777.
Bisection of A065608.
Sequence in context: A311046 A311047 A311048 * A311049 A073302 A191105
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jul 15 2022
STATUS
approved