Template:Sequence of the Day for November 24

Intended for: November 24, 2012

Timetable

• First draft entered by Alonso del Arte on November 26, 2011 ✓
• Draft (tentatively) reviewed (would someone please confirm the formula) by Daniel Forgues on November 26, 2011 ✓
• Draft to be approved by October 24, 2012

Thanksgiving 2016 falls on November 24.

Yesterday's SOTD * Tomorrow's SOTD

A002324: Number of divisors of

n

congruent to 1 modulo 3 minus number of divisors of

n

congruent to 2 modulo 3.

{ 1, 0, 1, 1, 0, 0, 2, 0, 1, 0, 0, 1, 2, ... }

These are the coefficients

a (n) = χ (n), n   ≥   1,

in expansion of Dirichlet series

L (s, χ )  :=    ∞ ∑   n  = 1    χ (n) n  s

=    ∞ ∏   i  = 1    1 (1 − (δ   pim + 1) pi  − s + δ   pim pi  − 2 s )

=      ∞ ∏   i  = 1    1 1 − δ   pim pi  − s     ∞ ∏   i  = 1    1 1 − pi  − s

=  1 1 − 3  − s     ∞ ∏   i  = 1    1 1 − pi  − s  , m = 3,

where

pi

is the

i

th prime.
Note:

m

was  − 3 in the formula, it makes more sense for it to be 3, would someone please confirm whether the formula is right.

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A000055: Number of trees with

n, n   ≥   0,

unlabeled nodes.

{ 1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, ... }

Trees with

n

unlabeled nodes

n	Trees
0	Null tree
1	●
2	●—●
3	●—●—●
4	● / ●—●—●—● ●—● \ ●
5	● ● / | ●—●—●—●—● ●—●—● ●—●—● \ | ● ●

This is the sequence for the example search on the front page of the OEIS. This Thanksgiving we are thankful for, among other things, the OEIS, which is an invaluable resource in many mathematical and scientific endeavors.

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