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%I #30 May 04 2021 15:50:22
%S 0,1,2,0,4,1,6,2,0,3,10,1,12,5,2,0,16,3,18,1,4,9,22,2,0,11,6,3,28,1,
%T 30,4,8,15,2,0,36,17,10,3,40,1,42,7,4,21,46,2,0,5,14,9,52,3,6,1,16,27,
%U 58,4,60,29,2,0,8,5,66,13,20,3,70,1,72,35,10,15,4
%N Minimum nonnegative integer m such that n = k*(k+m) for some positive integer k.
%C a(n) is difference between the least divisor of n that is >= square root(n) and the greatest divisor of n that is <= square root(n).
%C From _Omar E. Pol_, Aug 12 2009: (Start)
%C a(n) = 0 iff n is a square.
%C a(n) = n-1 is a new record iff n is a prime number. (End)
%C For odd n = 2k-1, a(n) = 2*A219695(k) is even. - _M. F. Hasler_, Nov 25 2012
%H T. D. Noe, <a href="/A056737/b056737.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = Min_{t - d | 0 < d <= t <= n and d*t=n}. - _Reinhard Zumkeller_, Feb 25 2002
%F a(n) = A033677(n)-A033676(n). - _Omar E. Pol_, Jun 21 2009
%F a(2n-1) = 2*A219695(n). - _M. F. Hasler_, Nov 25 2012
%e a(8) = 2 because 8 = 2*(2+2) and 8 = k*(k+1) or 8 = k^2 have no solutions for k = a positive integer.
%t A033676[n_] := If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2]], Sqrt[n]] A033677[n_] := If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2+1]], Sqrt[n]] Table[A033677[n] - A033676[n], {n, 1, 128}] (Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 27 2004)
%t Table[d = Divisors[n]; len = Length[d]; If[OddQ[len], 0, d[[1 + len/2]] - d[[len/2]]], {n, 100}] (* _T. D. Noe_, Jun 04 2012 *)
%o (PARI) A056737(n)={n=divisors(n);n[(2+#n)\2]-n[(1+#n)\2]} \\ _M. F. Hasler_, Nov 25 2012
%Y Cf. A033676, A033677.
%Y Cf. A000040, A000290, A147861, A163100, A163280.
%K nonn
%O 1,3
%A _Leroy Quet_, Aug 26 2000
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