A064727 Number of pairs x,y such that 0 < x <= y < n and x+y = n and x*y = kn for some k.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 3, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 2, 1, 0, 0, 0, 2, 4, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 3, 1, 5, 0, 0, 0, 1, 0

Offset

1,16

Comments

When m is an even square, a(m) is a record value in the sequence. [corrected by Jon E. Schoenfield, Nov 18 2017] and [corrected by Bernard Schott, Mar 08 2023]

Number of partitions of n into two parts (s,t) such that (s+t) | s*t. - Wesley Ivan Hurt, Apr 29 2021

From Bernard Schott, Mar 08 2023: (Start)

a(n) = 0 iff n is a squarefree number in A005117, so a(n) >= 1 when n is in A013929.

The least number m such that a(m) = t is m = A016742(t) = (2*t)^2. Example for t = 3, m = A016742(3) = 36 and the three corresponding partitions are (6,30), (12,24) and (18,18); so, these values of a(m) are also the records explained in 1st comment (see 2nd formula).

The least odd number m such that a(m) = u is m = A016754(u). (End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = Sum_{i=1..floor(n/2)} (1-ceiling(i*(n-i)/n)+floor(i*(n-i)/n). - Wesley Ivan Hurt, Apr 29 2021

a(A016742(n)) = a(A016754(n)) = n. - Bernard Schott, Mar 08 2023

Example

a(16) = 2 because 4+12 = 16 and 4*12 = 48 = 3*16, 8+8 = 16 and 8*8 = 4*16.

Mathematica

Table[Count[IntegerPartitions[n, {2}], _?(Divisible[Apply[Times, #], n] &)], {n, 105}] (* Michael De Vlieger, Nov 18 2017 *)

Prog

(PARI) A064727(n) = { my(s=0); for(x=1, n, y = (n-x); if((x<=y)&&(0==((x*y)%n)), s++)); (s); }; \\ Antti Karttunen, Nov 18 2017

Crossrefs

Cf. A005117, A013929, A016742, A016754.

Sequence in context: A366073 A362412 A071325 * A343222 A112378 A324832

Adjacent sequences: A064724 A064725 A064726 * A064728 A064729 A064730

Keyword

nonn,look

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 14 2001

Extensions

Offset corrected (to 1) by Antti Karttunen, Nov 18 2017

Status

approved