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%I #50 Jan 07 2025 20:26:17
%S 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,
%T 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,
%U 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
%N Number of pairs x,y such that 0 < x <= y < n and x+y = n and x*y = k*n for some k.
%C 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]
%C Number of partitions of n into two parts (s,t) such that (s+t) | s*t. - _Wesley Ivan Hurt_, Apr 29 2021
%C From _Bernard Schott_, Mar 08 2023: (Start)
%C a(n) = 0 iff n is a squarefree number in A005117, so a(n) >= 1 when n is in A013929.
%C 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).
%C The least odd number m such that a(m) = u is m = A016754(u). (End)
%H Antti Karttunen, <a href="/A064727/b064727.txt">Table of n, a(n) for n = 1..16384</a>
%F 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
%F a(A016742(n)) = a(A016754(n)) = n. - _Bernard Schott_, Mar 08 2023
%e a(16) = 2 because 4+12 = 16 and 4*12 = 48 = 3*16, 8+8 = 16 and 8*8 = 4*16.
%t Table[Count[IntegerPartitions[n, {2}], _?(Divisible[Apply[Times, #], n] &)], {n, 105}] (* _Michael De Vlieger_, Nov 18 2017 *)
%o (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
%Y Cf. A005117, A013929, A016742, A016754.
%K nonn,look
%O 1,16
%A Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 14 2001
%E Offset corrected (to 1) by _Antti Karttunen_, Nov 18 2017