A245212 a(n) = n * tau(n) - Sum_{(d<n) | n} (d * tau(d)).

1, 3, 5, 7, 9, 13, 13, 15, 20, 25, 21, 25, 25, 37, 43, 31, 33, 46, 37, 53, 63, 61, 45, 41, 64, 73, 74, 81, 57, 95, 61, 63, 103, 97, 115, 70, 73, 109, 123, 101, 81, 147, 85, 137, 166, 133, 93, 57, 132, 170, 163, 165, 105, 154, 187, 161, 183, 169, 117, 131, 121

Offset

1,2

Comments

If d are divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = (n * tau(n)) - Sum_{(d<n) | n} (d * tau(d)).

If a(n) = 0 then n must be > 10^7.

Conjecture: a(n) = sigma(n) iff n is a power of 2 (A000079).

Number n = 72 is the smallest number n such that a(n) < n (see A245213).

Number n = 144 is the smallest number n such that a(n) < 0 (see A245214).

Links

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Formula

a(n) = A038040(n) - A245211(n).

a(n) = 2 * A038040(n) - A060640(n) = 2 * (n * tau(n)) - Sum_{d | n} (d * tau(d)).

Example

For n = 6 with divisors [1, 2, 3, 6] we have: a(6) = 6 * tau(6) - (3 * tau(3) + 2 * tau(2) + 1 * tau(1)) = 6*4 - (3*2+2*2+1*1) = 13.

Prog

(Magma) [(2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])): n in [1..1000]];
(PARI) a(n) = sumdiv(n, d, (-1)^(d<n)*d*numdiv(d)) \\ Jens Kruse Andersen, Aug 13 2014

Crossrefs

Cf. A000005, A100484, A245211, A245213, A245214.

Sequence in context: A145341 A121388 A161700 * A063081 A337431 A067031

Adjacent sequences: A245209 A245210 A245211 * A245213 A245214 A245215

Keyword

sign

Author

Jaroslav Krizek, Jul 23 2014

Status

approved