A386935 Integers with the same arithmetic mean for divisors and anti-divisors.

3, 15, 135, 376, 6956, 1913646, 1838558856

Offset: 1

Author

Paolo P. Lava, Aug 09 2025

Keywords

• nonn
• hard
• more

Comments

For the listed numbers the arithmetic means are 2, 6, 30, 90, 1064, 97128, 143824680, ...

a(8) > 10^10, if it exists. - Amiram Eldar, Aug 12 2025

Examples

			Divisors of 135 are 8: 1, 3, 5, 9, 15, 27, 45, 135. Their sum is 240 and 240/8 = 30.
Anti-divisors of 135 are 7: 2, 6, 10, 18, 30, 54, 90. Their sum is 210 and 210/7 = 30.
		

Crossrefs

Cf. A000203/A000005, A066417/A066272.

Cf. A003601, A192284.

Programs

Maple

with(numtheory): P:=proc(q) local a, b, k, n, v; v:=[];
for n from 3 to q do k:=2; a:=0; b:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then b:=b+1; a:=a+k; fi; od;
if sigma(n)/tau(n)=a/b then v:=[op(v), n]; fi; od; op(v); end: P(10^4);

Python

from itertools import count, islice
from sympy.ntheory.factor_ import divisor_sigma, antidivisors
def A386935_gen(startvalue=3): # generator of terms >= startvalue
    for k in count(max(startvalue,3)):
        if divisor_sigma(k)*len(d:=antidivisors(k))==divisor_sigma(k,0)*sum(d):
            yield k
A386935_list = list(islice(A386935_gen(),5)) # Chai Wah Wu, Aug 12 2025

Extensions

a(6) from Michel Marcus, Aug 09 2025

a(7) from Amiram Eldar, Aug 10 2025