cp's OEIS Frontend

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

%I A217581 #59 Apr 25 2024 13:55:58
%S A217581 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,
%T A217581 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,
%U A217581 3,7,1,3,1,2,5,2,7,3,1,5,3,2,1,7,5,2,3
%N A217581 Largest prime divisor of n <= sqrt(n), 1 if n is prime or 1.
%C A217581 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
%H A217581 T. D. Noe, <a href="/A217581/b217581.txt">Table of n, a(n) for n = 1..10000</a>
%e A217581 From _Gus Wiseman_, Apr 06 2021: (Start)
%e A217581 The sequence selects the greatest element (or 1 if empty) of each of the following sets of strictly superior divisors:
%e A217581    1:{}     16:{2}      31:{}     46:{2}
%e A217581    2:{}     17:{}       32:{2}    47:{}
%e A217581    3:{}     18:{2,3}    33:{3}    48:{2,3}
%e A217581    4:{2}    19:{}       34:{2}    49:{7}
%e A217581    5:{}     20:{2}      35:{5}    50:{2,5}
%e A217581    6:{2}    21:{3}      36:{2,3}  51:{3}
%e A217581    7:{}     22:{2}      37:{}     52:{2}
%e A217581    8:{2}    23:{}       38:{2}    53:{}
%e A217581    9:{3}    24:{2,3}    39:{3}    54:{2,3}
%e A217581   10:{2}    25:{5}      40:{2,5}  55:{5}
%e A217581   11:{}     26:{2}      41:{}     56:{2,7}
%e A217581   12:{2,3}  27:{3}      42:{2,3}  57:{3}
%e A217581   13:{}     28:{2}      43:{}     58:{2}
%e A217581   14:{2}    29:{}       44:{2}    59:{}
%e A217581   15:{3}    30:{2,3,5}  45:{3,5}  60:{2,3,5}
%e A217581 (End)
%p A217581 A217581 := n -> `if`(isprime(n) or n=1, 1, max(op(select(i->i^2<=n, numtheory[factorset](n)))));
%t A217581 Table[If[n == 1 || PrimeQ[n], 1, Select[Transpose[FactorInteger[n]][[1]], # <= Sqrt[n] &][[-1]]], {n, 100}] (* _T. D. Noe_, Mar 25 2013 *)
%o A217581 (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
%Y A217581 Cf. A033676.
%Y A217581 Positions of first appearances are 1 and A001248.
%Y A217581 These divisors are counted by A063962.
%Y A217581 These divisors add up to A097974.
%Y A217581 The smallest prime factor of the same type is A107286.
%Y A217581 A strictly superior version is A341643.
%Y A217581 A superior version is A341676.
%Y A217581 A038548 counts superior (or inferior) divisors.
%Y A217581 A048098 lists numbers without a strictly superior prime divisor.
%Y A217581 A056924 counts strictly superior (or strictly inferior) divisors.
%Y A217581 A063538/A063539 have/lack a superior prime divisor.
%Y A217581 A140271 selects the smallest strictly superior divisor.
%Y A217581 A161906 lists inferior divisors.
%Y A217581 A207375 lists central divisors.
%Y A217581 A341591 counts superior prime divisors.
%Y A217581 A341642 counts strictly superior prime divisors.
%Y A217581 A341673 lists strictly superior divisors.
%Y A217581 - Inferior: A066839, A069288, A333749, A333750.
%Y A217581 - Superior: A033677, A051283, A059172, A070038, A116882, A116883, A161908, A341592, A341593, A341675.
%Y A217581 - Strictly Inferior: A060775, A333805, A333806, A341596, A341674.
%Y A217581 - Strictly Superior: A238535, A341594, A341595, A341644, A341645, A341646.
%Y A217581 Cf. A000005, A001055, A001221, A001222, A001248, A001414, A006530, A020639, A064052.
%K A217581 nonn
%O A217581 1,4
%A A217581 _Peter Luschny_, Mar 21 2013