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A211270
Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y = 2n.
8
0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 4, 1, 3, 3, 2, 1, 4, 2, 2, 3, 3, 1, 5, 1, 3, 3, 2, 3, 5, 1, 2, 3, 4, 1, 5, 1, 3, 5, 2, 1, 5, 2, 4, 3, 3, 1, 5, 3, 4, 3, 2, 1, 7, 1, 2, 5, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 2, 5, 3, 3, 5, 1, 5, 4, 2, 1, 7, 3, 2, 3, 4, 1, 8, 3, 3, 3, 2, 3, 6, 1, 4, 5
OFFSET
1,6
COMMENTS
For a guide to related sequences, see A211266.
LINKS
David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Theorem 4.2 p. 7.
FORMULA
a(n) = floor((A000005(2n)-1)/2). - Robert Israel, Feb 25 2019
EXAMPLE
a(12) counts these pairs: (2,12), (3,8), (4,6).
For n = 2, only the pair (2,2) satisfies the condition, thus a(2) = 1. - Antti Karttunen, Sep 30 2018
MAPLE
seq(floor((numtheory:-tau(2*n)-1)/2), n=1..100); # Robert Israel, Feb 25 2019
MATHEMATICA
a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}] (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}] (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}] (* this sequence *)
Table[c[n, 3*n], {n, 1, z1}] (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}] (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}] (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}] (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}] (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}] (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)
PROG
(PARI) A211270(n) = sumdiv(2*n, y, (((2*n/y)<=y)&&(y<=n))); \\ Antti Karttunen, Sep 30 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 07 2012
EXTENSIONS
Term a(2) corrected by Antti Karttunen, Sep 30 2018
STATUS
approved