A088949 -id:A088949 - cp's OEIS frontend

A088595 Numbers n such that (A006530(n) + A020639(n))/2 is an integer, divides n and it is not a power of prime number: it has at least 2 distinct prime factors. Special terms of A088948.

105, 231, 315, 525, 627, 693, 735, 897, 935, 945, 1155, 1575, 1581, 1617, 1729, 1881, 2079, 2205, 2465, 2541, 2625, 2691, 2835, 2967, 3135, 3465, 3525, 3675, 4123, 4301, 4389, 4485, 4675, 4715, 4725, 4743, 4851, 5145, 5487, 5643, 5775, 6237, 6279, 6545

Offset: 1

Labos Elemer, Nov 20 2003

nonn

Every number of the sequence has at least three different prime factors. Also, the sequence is infinite (it contains all numbers of the form 3^a*5^b*7^c with a,b,c>0). - Emmanuel Vantieghem, Nov 21 2016

			n = 315 = 3*3*5*7 is not a power of a prime, has 3 prime factors and (3+5)/2=7 divides n.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006530, A020639, A088948, A088949.

filter:= proc(n) local F;
  F:= numtheory:-factorset(n);
  nops(F) > 2 and n mod (min(F)+max(F))/2 = 0
end proc:
select(filter, [seq(i,i=1..10^4,2)]); # Robert Israel, Nov 21 2016

Rest@ Select[Range@ 6600, Function[n, And[IntegerQ@ #, Divisible[n, #], ! PrimePowerQ@ n] &[(#[[-1, 1]] + #[[1, 1]])/2] &@ FactorInteger@ n]] (* Michael De Vlieger, Nov 24 2016 *)

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.

Original entry on oeis.org

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

Labos Elemer, Nov 20 2003

nonn

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

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

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

Different from A046687.

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

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 *)

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

cp's OEIS Frontend

A088595 Numbers n such that (A006530(n) + A020639(n))/2 is an integer, divides n and it is not a power of prime number: it has at least 2 distinct prime factors. Special terms of A088948.

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