A319802 Even numbers without middle divisors.

10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 230, 232, 236, 244, 246, 248, 250, 254, 258, 262, 268, 274, 278, 282, 284

Offset: 1

Author

Omar E. Pol, Sep 28 2018

Keywords

• nonn

Comments

Even numbers k such that the symmetric representation of sigma(k) has an even number of parts.

For the definition of middle divisors, see A067742.

For more information about the symmetric representation of sigma(k) see A237593.

Let p be a prime > 5. Then a(n) is a number of the form m*p where m is an even number < sqrt(p). - David A. Corneth, Sep 28 2018

First differs from A244894 at a(51) = 230. - R. J. Mathar, Oct 04 2018

Is this twice A101550? - Omar E. Pol, Oct 04 2018

This sequence is not twice A101550: first differs at a(57) = 250 != 254 = 2*A101550(57). - Michael S. Branicky, Oct 14 2021

Examples

			10 is in the sequence because it's an even number and the symmetric representation of sigma(10) = 18 has an even number of parts as shown below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005843 and A071561.

Cf. A244894 (a subsequence).

Cf. A000203, A067742, A071090, A071562, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319796, A319801.

Programs

Python

from sympy import divisors
def ok(n):
    if n < 2 or n%2 == 1: return False
    return not any(n//2 <= d*d < 2*n for d in divisors(n, generator=True))
print(list(filter(ok, range(285)))) # Michael S. Branicky, Oct 14 2021