A192284 -id:A192284 - cp's OEIS frontend

A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.

1, 2, 11, 12, 15, 16, 22, 31, 76, 152, 309, 1576, 375479, 781314, 1114986, 3734218, 24311881, 68133239, 147881549

Offset: 1

Paolo P. Lava, Mar 13 2013

nonn

Tested up to k = 108122.

a(20) > 3*10^8. - Donovan Johnson, Mar 22 2013

			Anti-divisors of 76 are 3, 8, 9, 17 and 51 and their sum is 88.
Anti-divisor of 3 is 2 -> Sum is 2.
Anti-divisors of 8 are 3 and 5 -> Sum is 8.
Anti-divisors of 9 are 2 and 6 -> Sum is 8.
Anti-divisors of 17 are 2, 3, 5, 7 and 11 -> Sum is 28.
Anti-divisors of 51 are 2, 6 and 34 -> Sum is 42.
Finally, 2+8+8+28+42=88.

Cf. A066272, A178029, A191580, A191581, A192282-A192284.

A216213:= proc(q) local a,b,c,j,k,n;
for n from 1 to q do
  a:={}; b:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then b:=b+k; a:=a union {k}; fi; od;
  c:=0; for j from 1 to nops(a) do for k from 2 to a[j]-1 do if abs((a[j] mod k)-k/2)<1 then c:=c+k; fi; od; od; if b=c then print(n); fi; od; end:
A216213(10^10);

a(13)-a(19) from Donovan Johnson, Mar 22 2013

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

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Aug 09 2025

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

			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.

Cf. A000203/A000005, A066417/A066272.

Cf. A003601, A192284.

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);

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

a(6) from Michel Marcus, Aug 09 2025

a(7) from Amiram Eldar, Aug 10 2025

cp's OEIS Frontend

A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.

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

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cp's OEIS Frontend

A216213 Numbers k such that sigma*(k) = Sum_{j=anti-divisors of k} sigma*(j), where sigma*(k) is the sum of the anti-divisors of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Python

Extensions

A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.