A101322 a(n) = n - (least divisor of n greater than the square root of n) + (greatest divisor of n less than the square root of n) = n + A033676(n) - A033677(n).

1, 1, 1, 4, 1, 5, 1, 6, 9, 7, 1, 11, 1, 9, 13, 16, 1, 15, 1, 19, 17, 13, 1, 22, 25, 15, 21, 25, 1, 29, 1, 28, 25, 19, 33, 36, 1, 21, 29, 37, 1, 41, 1, 37, 41, 25, 1, 46, 49, 45, 37, 43, 1, 51, 49, 55, 41, 31, 1, 56, 1, 33, 61, 64, 57, 61, 1, 55, 49, 67, 1, 71, 1, 39, 65, 61, 73, 71, 1

Offset

1,4

Comments

a(n)/n represents, in some sense, how 'square' a positive integer n is. a(n)=1 iff n is a prime number. a(n)=n iff n is a square number. For nonsquare n, the first (note: not zeroth) partial quotient of the continued fraction of a(n)/n is n iff n is prime, else 1.

Links

Table of n, a(n) for n=1..79.

Example

a(6) = 5 because 6-3+2=5
a(7) = 1 because 7-7+1=1

Mathematica

Table[n - If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2 + 1]], Sqrt[n]] + If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2]], Sqrt[n]], {n, 1, 128}]

Crossrefs

Cf. A033676, A033677, A000040, A000290.

Sequence in context: A357311 A107463 A157104 * A029644 A345305 A024919

Adjacent sequences: A101319 A101320 A101321 * A101323 A101324 A101325

Keyword

base,nonn

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 24 2004

Status

approved