A088948 - OEIS

%I #26 Mar 02 2023 13:50:11

%S 2,3,4,5,7,8,9,11,13,15,16,17,19,21,23,25,27,29,31,32,33,35,37,39,41,

%T 43,45,47,49,51,53,55,57,59,61,63,64,65,67,69,71,73,75,77,79,81,83,85,

%U 87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125

%N Numbers k such that (A006530(k) + A020639(k))/2 is an integer; that is, arithmetic mean of least and largest prime factor is an integer.

%C Union of odd numbers and powers of 2 minus {1}. - _Ivan Neretin_, Dec 30 2015

%C In other words, the symmetric difference of sets A005408 (all prime factors are odd) and A000079 (all prime factors are even). If we had allowed 1 as a member, it would have been the union of A005408 and A000079, as stated. - _Jeppe Stig Nielsen_, Dec 27 2019

%H Michael De Vlieger, <a href="/A088948/b088948.txt">Table of n, a(n) for n = 1..10000</a>

%e Primes and prime powers are here.

%e Also other composites: n=105, (3+7)/2 = 5 is an integer (and, moreover, divides n).

%t Rest@ Select[Range@ 125, IntegerQ[(FactorInteger[#][[1, 1]] + FactorInteger[#][[-1, 1]])/2] &] (* _Michael De Vlieger_, Mar 28 2015 *)

%t amintQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},IntegerQ[Mean[{fi[[1]],fi[[-1]]}]]]; Select[Range[150],amintQ] (* _Harvey P. Dale_, Mar 02 2023 *)

%o (PARI) is_a088948(n) = {local (f);f=factor(n);if(Mod(vecmin(f[,1])+vecmax(f[,1]),2)==0,1,0)} \\ _Michael B. Porter_, Mar 28 2015

%Y Different from A046687.

%Y Cf. A006530, A020639, A088949, A088949, A046687.

%K nonn

%O 1,1

%A _Labos Elemer_, Nov 20 2003