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

1,3

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

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

Example

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.

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, 128}] (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 *)

Prog

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

Crossrefs

Cf. A033676, A033677.

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

Sequence in context: A243981 A369657 A338824 * A289144 A350576 A008797

Adjacent sequences: A056734 A056735 A056736 * A056738 A056739 A056740

Keyword

nonn

Author

Leroy Quet, Aug 26 2000

Status

approved