A101322 a(n) = n - (least divisor of n >= the square root of n) + (greatest divisor of n <= the square root of 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, 78, 81, 43, 1, 79, 73, 45, 61, 85, 1, 89, 85

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 (or 1). 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

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = n + A033676(n) - A033677(n).

Example

a(6) = 5 because 6-3+2=5
a(7) = 1 because 7-7+1=1
a(9) = 9 because 9-3+3=9.

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}]

Prog

(PARI) A101322(n) = fordiv(n, d, if((d^2) >= n, return(n+(n/d)-d))); \\ Antti Karttunen, Jan 18 2025

Crossrefs

Cf. A008578 (positions of 1's), A000290 (fixed points), A033676, A033677.

Sequence in context: A373364 A107463 A157104 * A029644 A345305 A024919

Adjacent sequences: A101319 A101320 A101321 * A101323 A101324 A101325

Keyword

nonn

Author

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

Extensions

Name corrected to match the given formula and the data, more terms added by Antti Karttunen, Jan 18 2025

Status

approved