A340805 a(n) is the number of solutions of the equation x*(y - 1) + (2*x - y - 1)*(x mod 2) = 2*n for 0 < x <= y.

1, 1, 1, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 5, 4, 4, 2, 7, 2, 4, 4, 6, 2, 6, 2, 6, 5, 4, 2, 8, 2, 5, 4, 6, 2, 6, 3, 6, 4, 4, 2, 10, 2, 4, 4, 6, 4, 7, 2, 6, 4, 6, 2, 9, 2, 4, 6, 6, 2, 7, 2, 8, 4, 4, 2, 10, 4, 4, 4

Offset: 1

Author

Stefano Spezia, Jan 22 2021

Keywords

• nonn

Comments

Also the number of times 2*n+1 appears in A340804.

Offset is 1 because the equation x*(y - 1) + (2*x - y - 1)*(x mod 2) = 0 has an infinite number of positive integer solutions satisfying the inequality x <= y, or equivalently, A005408(0) = 1 appears infinitely many times in A340804.

Examples

			a(6) = 3 since there are 3 positive integer solutions (x, y) satisfying the inequality x <= y, i.e., (2, 4), (3, 5) and (4, 7).
		

Links

• Stefano Spezia, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A005408 (2*n+1), A038548, A340804.

Programs

Mathematica

Table[Sum[Boole[d<=Floor[(1+Sqrt[1+8n])/4]]+Boole[d<=Floor[(Sqrt[1+2n]-1)/2]],{d,Divisors[n]}],{n,87}]

PARI

T(n, k) = 1 + k*(n - 1) + (2*k - n - 1)*(k % 2); \\ A340804
a(n) = sum(i=1, 2*n+1, sum(k=1, i, T(i, k) == 2*n+1)); \\ Michel Marcus, Jan 25 2021

PARI

a(n) = sumdiv(n, d, (d <= (1 + sqrt(1 + 8*n))\4) + (d <= (sqrt(1 + 2*n) - 1)\2)); \\ Michel Marcus, Jan 25 2021

Formula

a(n) = Sum_{d|n} ([d <= floor((1 + sqrt(1 + 8*n))/4)] + [d <= floor((sqrt(1 + 2*n) - 1)/2)]), where [ ] is the Iverson bracket.

A038548(n) <= a(n) <= A000005(n).

a(p) = A000005(p) = 2 if p is a prime greater than or equal to 5.