A096157 Numbers whose numbers of odd and even proper divisors differ at most by 1.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 20, 22, 23, 26, 28, 29, 30, 31, 34, 37, 38, 41, 42, 43, 44, 46, 47, 50, 52, 53, 54, 58, 59, 61, 62, 66, 67, 68, 70, 71, 73, 74, 76, 78, 79, 82, 83, 86, 89, 90, 92, 94, 97, 98, 101, 102, 103, 106, 107, 109, 110, 113

Offset: 1

Author

Reinhard Zumkeller, Jun 18 2004

Keywords

• nonn

Comments

A union of 1, 4, primes, quadrupled primes, and all numbers of the form 4k+2. - Ivan Neretin, Aug 12 2015

Examples

			Proper divisors of 42 are {1,2,3,6,7,14,21}: 4=#{1,3,7,21} are odd and 3=#{2,6,14} are even, as abs(4-3)=1, 42 is a term.
		

Links

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

Crossrefs

Cf. A032741, A091954.

Programs

Maple

filter:= n -> if n::odd then isprime(n) elif n mod 4 = 0 then isprime(n/4) else true fi:
filter(1):= true: filter(4):= true:
select(filter, [$1..1000]); # Robert Israel, Aug 12 2015

Mathematica

okQ[n_] := With[{dd = Most[Divisors[n]]},
     Abs[2 Count[dd, _?OddQ] - Length[dd]] <= 1];
Select[Range[1000], okQ] (* Jean-François Alcover, Dec 16 2021 *)

PARI

is(n)=if(bittest(n,0),isprime(n),bittest(n,1)||isprime(n>>2))||n<9 \\ M. F. Hasler, Aug 13 2015

Formula

Abs(A032741(a(n)) - 2*A091954(a(n))) <= 1.