A339010 a(n) is the number of ways to write n as the difference of two centered k-gonal numbers for k >= 3.

0, 0, 1, 1, 1, 2, 1, 2, 3, 2, 1, 5, 1, 2, 5, 3, 1, 6, 1, 5, 5, 2, 1, 8, 3, 2, 6, 5, 1, 10, 1, 4, 5, 2, 5, 12, 1, 2, 5, 8, 1, 10, 1, 5, 12, 2, 1, 11, 3, 6, 5, 5, 1, 12, 5, 8, 5, 2, 1, 19, 1, 2, 12, 5, 5, 10, 1, 5, 5, 10, 1, 18, 1, 2, 12, 5, 5, 10, 1, 11, 10, 2

Offset

1,6

Comments

Records occur at indices n = 1, 3, 6, 9, 12, 18, 24, 30, 36, 60, 90, 120, 180, 270, 360, 420, 540, 630, 840, 1080, ...

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Code Golf Stack Exchange, Uncentered Polygons

OEIS Wiki, Centered polygonal numbers

Eric Weisstein's World of Mathematics, Centered Polygonal Number

Wikipedia, Centered polygonal number

Formula

a(n) = Sum_{d|n, 3*d <= n} A001227(d).

Example

For n = 35, the a(35) = 5 differences are:
A101321( 5,4) - A101321( 5,2) =  51 -  16 = 35,
A101321( 5,7) - A101321( 5,6) = 141 - 106 = 35,
A101321( 7,3) - A101321( 7,1) =  43 -   8 = 35,
A101321( 7,5) - A101321( 7,4) = 106 -  71 = 35, and
A101321(36,1) - A101321(36,0) =  36 -   1 = 35.

Prog

(PARI) a(n) = sumdiv(n, d, if (3*d <= n, numdiv(d>>valuation(d, 2)))); \\ Michel Marcus, Nov 19 2020

Crossrefs

Cf. A001227, A101321.

Cf. A333822 (polygonal numbers), A333836 (positive polygonal numbers), A333868 (binomial coefficients), A333880 (perfect powers).

Sequence in context: A332509 A336157 A190167 * A332842 A352696 A171565

Adjacent sequences: A339007 A339008 A339009 * A339011 A339012 A339013

Keyword

nonn

Author

Peter Kagey, Nov 18 2020

Status

approved