A333822 -id:A333822 - cp's OEIS frontend

A333836 Number of ways to write n as the difference of two positive k-gonal numbers for k >= 3.

0, 1, 2, 2, 4, 2, 5, 3, 6, 3, 6, 4, 7, 4, 7, 5, 8, 4, 7, 5, 10, 6, 7, 5, 10, 5, 10, 5, 9, 7, 9, 6, 11, 6, 10, 6, 12, 5, 11, 7, 11, 6, 9, 7, 13, 9, 9, 8, 12, 7, 13, 7, 9, 7, 11, 9, 17, 7, 7, 8, 13, 6, 14, 9, 17, 8, 11, 6, 12, 9, 11, 9, 13, 7

Offset: 1

Peter Kagey, Apr 07 2020

nonn

Records occur at indices 1, 2, 3, 5, 7, 9, 13, 17, 21, 33, 37, 45, 57, 105, 145, 217, 225, 273, 385, 495, 561, 651, 705, 945, 1105, ... - Peter Kagey, Nov 18 2020

			The a(9) = 6 ways of writing 9 as the difference of two k-gonal numbers are:
A000217(4) - A000217(1) = 10 -  1 (triangular),
A000217(5) - A000217(3) = 15 -  6 (triangular),
A000217(9) - A000217(8) = 45 - 36 (triangular),
A000290(5) - A000290(4) = 25 - 16 (square),
A000384(3) - A000384(2) = 15 -  6 (hexagonal), and
A001107(2) - A001107(1) = 10 -  1 (10-gonal).

Peter Kagey, Table of n, a(n) for n = 1..5000
OEIS Wiki, Figurate numbers: Polygonal numbers

Cf. A177025, A333822.

Cf. A000217, A000290, A000384, A001107.

b := 74
CoefficientList[
Series[Sum[
   Sum[x^(k*(p*k - (p - 2))/2)*x^(p*k)/(1 - x^(p*k)), {k, 1, b}], {p,
    1, b - 1}], {x, 0, b}], x]

a(n) = A333822(n) - A177025(n) for n > 2.

A333868 The number of ways to write n as the difference of two k-simplex numbers for k >= 2.

Original entry on oeis.org

1, 3, 3, 4, 5, 4, 3, 6, 7, 4, 4, 4, 5, 9, 4, 4, 5, 5, 7, 9, 4, 4, 4, 6, 4, 7, 7, 4, 7, 5, 3, 6, 6, 11, 9, 4, 4, 6, 4, 4, 6, 4, 5, 11, 5, 4, 4, 6, 6, 6, 5, 4, 7, 12, 8, 6, 4, 4, 6, 4, 4, 8, 5, 8, 9, 4, 4, 7, 8, 4, 5, 4, 5, 8, 4, 8, 9, 4, 5, 8, 4, 6, 10, 7, 4, 6

Offset: 2

Peter Kagey, Apr 08 2020

nonn

a(n) >= A001227(n) + A307666(n).
a(n) >= A003016(n) + A003016(n+1) - 2.
Records occur at indices 2, 3, 5, 6, 9, 10, 15, 35, 55, 105, 210, 1365, 2925, 3003,...

			The a(9) = 6 ways to write 9 as the difference of k-simplex numbers for k > 2 are:
C(5,  2) - C(2, 2) = 10 -  1,
C(6,  2) - C(4, 2) = 15 -  6,
C(10, 2) - C(9, 2) = 45 - 36,
C(5,  3) - C(3, 3) = 10 -  1,
C(9,  8) - C(7, 8) =  9 -  0, and
C(10, 9) - C(9, 9) = 10 -  1,
where C(n,k) = binomial(n,k) = A007318(n,k).

Peter Kagey, Table of n, a(n) for n = 2..5000

Cf. A001227, A003016, A007318, A307666.
Cf. A333822, A333836.
The k-simplex numbers for 2 <= k <= 6 are A000217 (k=2), A000292 (k=3), A000332 (k=4), A000389 (k=5), and A000579 (k=6).

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

Original entry on oeis.org

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

A333836 Number of ways to write n as the difference of two positive k-gonal numbers for k >= 3.

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A333868 The number of ways to write n as the difference of two k-simplex numbers for k >= 2.

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

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

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