A217581 Largest prime divisor of n <= sqrt(n), 1 if n is prime or 1.

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 5, 2, 3, 2, 1, 5, 1, 2, 3, 2, 5, 3, 1, 2, 3, 5, 1, 3, 1, 2, 5, 2, 1, 3, 7, 5, 3, 2, 1, 3, 5, 7, 3, 2, 1, 5, 1, 2, 7, 2, 5, 3, 1, 2, 3, 7, 1, 3, 1, 2, 5, 2, 7, 3, 1, 5, 3, 2, 1, 7, 5, 2, 3

Offset: 1

Author

Peter Luschny, Mar 21 2013

Keywords

• nonn

Comments

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by A161906. This sequence selects the greatest inferior prime divisor of n. - Gus Wiseman, Apr 06 2021

Examples

			From _Gus Wiseman_, Apr 06 2021: (Start)
The sequence selects the greatest element (or 1 if empty) of each of the following sets of strictly superior divisors:
   1:{}     16:{2}      31:{}     46:{2}
   2:{}     17:{}       32:{2}    47:{}
   3:{}     18:{2,3}    33:{3}    48:{2,3}
   4:{2}    19:{}       34:{2}    49:{7}
   5:{}     20:{2}      35:{5}    50:{2,5}
   6:{2}    21:{3}      36:{2,3}  51:{3}
   7:{}     22:{2}      37:{}     52:{2}
   8:{2}    23:{}       38:{2}    53:{}
   9:{3}    24:{2,3}    39:{3}    54:{2,3}
  10:{2}    25:{5}      40:{2,5}  55:{5}
  11:{}     26:{2}      41:{}     56:{2,7}
  12:{2,3}  27:{3}      42:{2,3}  57:{3}
  13:{}     28:{2}      43:{}     58:{2}
  14:{2}    29:{}       44:{2}    59:{}
  15:{3}    30:{2,3,5}  45:{3,5}  60:{2,3,5}
(End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A033676.

Positions of first appearances are 1 and A001248.

These divisors are counted by A063962.

These divisors add up to A097974.

The smallest prime factor of the same type is A107286.

A strictly superior version is A341643.

A superior version is A341676.

A038548 counts superior (or inferior) divisors.

A048098 lists numbers without a strictly superior prime divisor.

A056924 counts strictly superior (or strictly inferior) divisors.

A063538/A063539 have/lack a superior prime divisor.

A140271 selects the smallest strictly superior divisor.

A161906 lists inferior divisors.

A207375 lists central divisors.

A341591 counts superior prime divisors.

A341642 counts strictly superior prime divisors.

A341673 lists strictly superior divisors.

- Inferior: A066839, A069288, A333749, A333750.

- Superior: A033677, A051283, A059172, A070038, A116882, A116883, A161908, A341592, A341593, A341675.

- Strictly Inferior: A060775, A333805, A333806, A341596, A341674.

- Strictly Superior: A238535, A341594, A341595, A341644, A341645, A341646.

Cf. A000005, A001055, A001221, A001222, A001248, A001414, A006530, A020639, A064052.

Programs

Maple

A217581 := n -> `if`(isprime(n) or n=1, 1, max(op(select(i->i^2<=n, numtheory[factorset](n)))));

Mathematica

Table[If[n == 1 || PrimeQ[n], 1, Select[Transpose[FactorInteger[n]][[1]], # <= Sqrt[n] &][[-1]]], {n, 100}] (* T. D. Noe, Mar 25 2013 *)

PARI

a(n) = {my(m=1); foreach(factor(n)[,1], d, if(d^2 <= n, m=max(m,d))); m} \\ Andrew Howroyd, Oct 11 2023