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

1,2

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.

Formula

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

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

Example

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.

Maple

f:= proc(n) local F, F2, m, i;
    F:= sort(ifactors(n)[2], (i, j) -> i[1]<j[1]);
    F2:= ListTools:-PartialSums(map2(op, 2, F));
    for i from 1 do
      if 2*F2[i]>=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 *)

Prog

(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

Crossrefs

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

Sequence in context: A325643 A353270 A356838 * A071889 A091963 A067695

Adjacent sequences: A079876 A079877 A079878 * A079880 A079881 A079882

Keyword

nonn

Author

Reinhard Zumkeller, Jan 13 2003

Extensions

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

Status

approved