A079879 Median prime factor: a(1)=1 and for n>1: least prime p such that not more than floor(Omega(n)/2) prime factors are greater than p; Omega(n) is the total number of prime factors of n (A001222).

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 3, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 3, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 3, 43, 2, 3, 2, 47, 2, 7, 5, 3, 2, 53, 3, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 3, 67, 2, 3, 5, 71, 2, 73, 2, 5, 2, 7, 3, 79, 2, 3, 2, 83, 2, 5, 2, 3, 2, 89, 3, 7, 2, 3, 2, 5, 2

Offset: 1

Author

Reinhard Zumkeller, Jan 13 2003

Keywords

• nonn

Examples

			a(30)=a(2*3*5)=3; a(60)=a(2*2*3*5)=2; a(72)=a(2*2*2*3*3)=2; a(90)=a(2*3*3*5)=3; a(108)=a(2*2*3*3*3)=3; a(144)=a(2*2*2*2*3*3)=2; a(216)=a(2*2*2*3*3*3)=2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Jean-Marie De Koninck and Florian Luca, On the Middle Prime Factor of an Integer, Journal of Integer Sequences, Vol. 16 (2013), Article 13.5.5.
• Nicolas Doyon and Vincent Ouellet, On the sum of the reciprocals of the middle prime factor counting multiplicity, Annales Univ. Sci. Budapest., Sect. Comp. 47 (2018) 249-259.

Crossrefs

Cf. A027746, A033676, A001222, A079880, A079881.

Programs

Maple

f:= proc(n) local F,F2,m,i;
    F:= sort(ifactors(n)[2],(i,j) -> i[1]=F2[-1] then return F[i][1] fi
    od
end proc:
1, seq(f(n), n=2..100); # Robert Israel, Aug 25 2015

Mathematica

f[n_] := Block[{p = Flatten[Table[#1, {#2}] & @@@ FactorInteger@ n], len}, len = Length@ p; If[OddQ@ len, p[[1 + Floor[len/2]]], p[[len/2]]]]; {1}~Join~Table[f@ n, {n, 2, 96}] (* Michael De Vlieger, Aug 25 2015 *)

PARI

a(n) = {if (n==1, return(1)); my(f=factor(n), v=vector(bigomega(f)), j=1); for (k=1, #f~, for (i=1, f[k,2], v[j]=f[k,1]; j++);); v[(#v+1)\2];} \\ Michel Marcus, Apr 15 2022

Formula

A020639(n) <= a(n) <= A006530(n);

a(m) = A020639(m) = A006530(m) iff m is a prime power (A000961).

Extensions

Typo fixed by Franklin T. Adams-Watters, Jul 10 2012