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A064727
Number of pairs x,y such that 0 < x <= y < n and x+y = n and x*y = kn for some k.
1
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
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
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