A308644 Number of ways to write n as (3^a*5^b)^2 + c*(3c+1)/2 + d*(7d+1)/2, where a and b are nonnegative integers, and c and d are integers.

1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 3, 4, 2, 5, 3, 2, 3, 2, 3, 1, 3, 4, 3, 4, 5, 3, 5, 4, 6, 2, 2, 4, 4, 6, 2, 4, 6, 7, 5, 3, 4, 6, 3, 4, 4, 2, 4, 3, 4, 3, 3, 4, 5, 5, 5, 2, 3, 8, 3, 5, 4, 7, 5, 4, 4, 4, 4, 5, 4, 1, 4, 5, 4, 1, 3, 3, 6, 4, 7, 7, 3, 5, 7, 8, 2, 4, 5, 6, 7, 3, 8, 5, 7, 8, 4, 7, 8, 2

Offset

1,5

Comments

Conjecture 1: a(n) > 0 for all n > 0. Also, any positive integer n can be written as (3^a*5^b)^2 + c*(3c+1)/2 + d*(7d+3)/2, where a and b are nonnegative integers, and c and d are integers.

Conjecture 2: Let r be 1 or 3. Then, any positive integer n can be written as (3^a*4^b)^2 + c*(3c+1)/2 + d*(7d+r)/2, where a and b are nonnegative integers, and c and d are integers.

We have verified Conjectures 1-2 for all n = 1..10^6.

See also A308640, A308641 and A308656 for similar conjectures.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math., in press.

Example

a(152) = 1 with 152 = (3^0*5^0)^2 + (-4)*(3*(-4)+1)/2 + 6*(7*6+1)/2.
a(129894) = 1 with 129894 = (3^0*5^1)^2 + 154*(3*154+1)/2 + 164*(7*164+1)/2.
a(200963) = 1 with 200963 = (3^1*5^0)^2 + 364*(3*364+1)/2 + 24*(7*24+1)/2.
a(371278) = 1 with 371278 = (3^3*5^1)^2 + (-382)*(3*(-382)+1)/2 + (-196)*(7*(-196)+1)/2.
a(534699) = 1 with 534699 = (3^2*5^2)^2 + 543*(3*543+1)/2 + (-109)*(3*(-109)+1)/2.

Mathematica

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
tab={}; Do[r=0; Do[If[PenQ[n-9^a*25^b-x(7x+1)/2], r=r+1], {a, 0, Log[9, n]}, {b, 0, Log[25, n/9^a]}, {x, -Floor[(Sqrt[56(n-9^a*25^b)+1]+1)/14], (Sqrt[56(n-9^a*25^b)+1]-1)/14}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000244, A000290, A000351, A001318, A308566, A308584, A308621, A308623, A308640, A308641, A308656.

Sequence in context: A070913 A266743 A284050 * A308184 A114280 A123564

Adjacent sequences: A308641 A308642 A308643 * A308645 A308646 A308647

Keyword

nonn

Author

Zhi-Wei Sun, Jun 13 2019

Status

approved