A056737 Minimum nonnegative integer m such that n = k*(k+m) for some positive integer k.

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, 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, 58, 4, 60, 29, 2, 0, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 10, 15, 4

Offset: 1

Author

Leroy Quet, Aug 26 2000

Keywords

• nonn

Comments

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).

From Omar E. Pol, Aug 12 2009: (Start)

a(n) = 0 iff n is a square.

a(n) = n-1 is a new record iff n is a prime number. (End)

For odd n = 2k-1, a(n) = 2*A219695(k) is even. - M. F. Hasler, Nov 25 2012

Conjecture: There exists some constant, k, approximately equal to 1.7, such that a(n) is of average order k*n/log(n). See Tooth Link for evidence. - Clive Tooth, Mar 18 2025

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Clive Tooth, [Sum_{i=2..n} a(i)]/[Sum_{i=2..n} i/log(i)] for n=10^6 to 10^8

Crossrefs

Cf. A033676, A033677.

Cf. A000040, A000290, A147861, A163100, A163280.

Programs

Mathematica

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, 77}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 27 2004 *)
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 *)

PARI

A056737(n)={n=divisors(n);n[(2+#n)\2]-n[(1+#n)\2]}  \\ M. F. Hasler, Nov 25 2012

Formula

a(n) = Min_{t - d | 0 < d <= t <= n and d*t=n}. - Reinhard Zumkeller, Feb 25 2002

a(n) = A033677(n)-A033676(n). - Omar E. Pol, Jun 21 2009

a(2n-1) = 2*A219695(n). - M. F. Hasler, Nov 25 2012