A088948 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.

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, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 1

Author

Labos Elemer, Nov 20 2003

Keywords

• nonn

Comments

Union of odd numbers and powers of 2 minus {1}. - Ivan Neretin, Dec 30 2015

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

Examples

			Primes and prime powers are here.
Also other composites: n=105, (3+7)/2 = 5 is an integer (and, moreover, divides n).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Different from A046687.

Cf. A006530, A020639, A088949, A088949, A046687.

Programs

Mathematica

Rest@ Select[Range@ 125, IntegerQ[(FactorInteger[#][[1, 1]] + FactorInteger[#][[-1, 1]])/2] &] (* Michael De Vlieger, Mar 28 2015 *)
amintQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},IntegerQ[Mean[{fi[[1]],fi[[-1]]}]]]; Select[Range[150],amintQ] (* Harvey P. Dale, Mar 02 2023 *)

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