A245211 a(n) = Sum_{(d<n) | n} (d * tau(d)).

0, 1, 1, 5, 1, 11, 1, 17, 7, 15, 1, 47, 1, 19, 17, 49, 1, 62, 1, 67, 21, 27, 1, 151, 11, 31, 34, 87, 1, 145, 1, 129, 29, 39, 25, 254, 1, 43, 33, 219, 1, 189, 1, 127, 104, 51, 1, 423, 15, 130, 41, 147, 1, 278, 33, 287, 45, 63, 1, 589, 1, 67, 132, 321, 37, 277

Offset

1,4

Comments

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

Number n = 144 is the smallest number n such that a(n) > n * tau(n) (see A245212 and A245214).

Conjecture: 21 is only number such that a(n) = n.

Links

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

Formula

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

a(n) = A038040(n) - A245212(n).

a(n) = 1 for n = primes.

a(n) = n + 5 for even semiprimes q = 2p > 4 (see A100484) where p = odd prime.

Example

For n = 21 with proper divisors [1, 3, 7] we have: a(21) = 7 * tau(7) + 3 * tau(3) + 1 * tau(1) = 7*2 + 3*2 + 1*1 = 21.

Prog

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

Crossrefs

Cf. A000005, A100484, A245212, A245213, A245214.

Sequence in context: A189745 A394625 A062967 * A356775 A347389 A330774

Adjacent sequences: A245208 A245209 A245210 * A245212 A245213 A245214

Keyword

nonn

Author

Jaroslav Krizek, Jul 23 2014

Status

approved