A333868 -id:A333868 - cp's OEIS frontend

A333880 Number of integer solutions to n = x^k - y^k with x > y >= 0 and k >= 2.

0, 1, 1, 1, 0, 2, 2, 2, 0, 1, 1, 1, 0, 3, 3, 1, 0, 2, 1, 2, 0, 1, 2, 2, 1, 3, 1, 1, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 2, 1, 0, 1, 1, 3, 0, 1, 3, 2, 0, 2, 1, 1, 0, 2, 3, 2, 0, 1, 2, 2, 0, 5, 5, 3, 0, 1, 1, 2, 0, 1, 3, 1, 0, 3, 1, 2, 0, 1, 4, 4, 0, 1, 2, 2, 0, 2, 2

Offset: 2

Peter Kagey, Apr 08 2020

nonn

a(n) = 0 for all n in A303744.

Records occur at n = 2, 3, 7, 15, 63, 240, 480, 720, 960, 1440, 2400, 2880, 3360, 4032, 5040, ... - Peter Kagey, Nov 18 2020

			For n = 63, the a(63) = 5 solutions are
   8^2 -  1^2 =   64 -   1,
  12^2 -  9^2 =  144 -  81,
  32^2 - 31^2 = 1024 - 961,
   4^3 -  1^3 =   64 -   1, and
   2^6 -  1^6 =   64 -   1.

Peter Kagey, Table of n, a(n) for n = 2..10000

Cf. A034178, A303744, A333822, A333868.

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

Original entry on oeis.org

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

Peter Kagey, Nov 18 2020

nonn

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, ...

			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.

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

Cf. A001227, A101321.

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

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

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

cp's OEIS Frontend

A333880 Number of integer solutions to n = x^k - y^k with x > y >= 0 and k >= 2.

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