A340804 -id:A340804 - cp's OEIS frontend

A340805 a(n) is the number of solutions of the equation x(y - 1) + (2x - y - 1)(x mod 2) = 2n 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

Stefano Spezia, Jan 22 2021

nonn

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.

			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).

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

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

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

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

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

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.

A341829 Irregular triangle read by rows: the n-th row gives the x-values of the solutions of the equation x(y - 1) + (2x - y - 1)(x mod 2) = 2n for 0 < x <= y.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Feb 21 2021

Equivalently, the n-th row gives the column indices corresponding to 2*n + 1 in the triangle A340804.

			Triangle begins:
2
2
2
2   3
2   3
2   3   4
2   3
2   3   4
2   3
2   3   4
2   3
2   3   4   5
2   3
2   3   4   5
2   3   6
2   3   4   5
2   3
2   3   4   5   6
...

Stefano Spezia, Table of n, a(n) for n = 1..10175 (first 1500 rows of the triangle, flattened).

Cf. A005843, A340804, A340805 (row length or solutions number), A341830 (y-values).

Table[Union[2Intersection[Divisors[n],Table[d,{d,Floor[(1+Sqrt[1+8n])/4]}]],2Intersection[Divisors[n],Table[d,{d,Floor[(Sqrt[1+2n]-1)/2]}]]+1],{n,30}]//Flatten

A341830 Irregular triangle read by rows: the n-th row gives the y-values of the solutions of the equation x(y - 1) + (2x - y - 1)(x mod 2) = 2n for 0 < x <= y.

Original entry on oeis.org

2, 3, 4, 3, 5, 4, 6, 4, 5, 7, 6, 8, 5, 7, 9, 8, 10, 6, 9, 11, 10, 12, 5, 7, 11, 13, 12, 14, 6, 8, 13, 15, 6, 14, 16, 7, 9, 15, 17, 16, 18, 7, 8, 10, 17, 19, 18, 20, 9, 11, 19, 21, 8, 20, 22, 10, 12, 21, 23, 22, 24, 7, 9, 11, 13, 23, 25, 24, 26, 12, 14, 25, 27, 8, 10, 26, 28

Offset: 1

Stefano Spezia, Feb 21 2021

Equivalently, the n-th row gives the row indices corresponding to 2*n + 1 in the triangle A340804.

			Triangle begins:
   2
   3
   4
   3   5
   4   6
   4   5   7
   6   8
   5   7   9
   8  10
   6   9  11
  10  12
   5   7  11  13
  12  14
   6   8  13  15
   6  14  16
   7   9  15  17
  16  18
   7   8  10  17  19
  ...

Stefano Spezia, Table of n, a(n) for n = 1..10175 (first 1500 rows of the triangle, flattened).

Cf. A005843, A340804, A340805 (row length or solutions number), A341829 (x-values).

Table[Union[n/Intersection[Divisors[n],Table[d,{d,Floor[(1+Sqrt[1+8n])/4]}]]+1,n/Intersection[Divisors[n],Table[d,{d,Floor[(Sqrt[1+2n]-1)/2]}]]-1],{n,27}]//Flatten

cp's OEIS Frontend

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

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A341829 Irregular triangle read by rows: the n-th row gives the x-values of the solutions of the equation x(y - 1) + (2x - y - 1)(x mod 2) = 2n for 0 < x <= y.

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A341830 Irregular triangle read by rows: the n-th row gives the y-values of the solutions of the equation x(y - 1) + (2x - y - 1)(x mod 2) = 2n for 0 < x <= y.

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cp's OEIS Frontend

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.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

PARI

Formula

A341829 Irregular triangle read by rows: the n-th row gives the x-values of the solutions of the equation x*(y - 1) + (2*x - y - 1)*(x mod 2) = 2*n for 0 < x <= y.

Views

Author

Keywords

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Programs

Mathematica

A341830 Irregular triangle read by rows: the n-th row gives the y-values of the solutions of the equation x*(y - 1) + (2*x - y - 1)*(x mod 2) = 2*n for 0 < x <= y.

Views

Author

Keywords

Comments

Examples

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Mathematica

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

A341829 Irregular triangle read by rows: the n-th row gives the x-values of the solutions of the equation x(y - 1) + (2x - y - 1)(x mod 2) = 2n for 0 < x <= y.

A341830 Irregular triangle read by rows: the n-th row gives the y-values of the solutions of the equation x(y - 1) + (2x - y - 1)(x mod 2) = 2n for 0 < x <= y.