A191580 -id:A191580 - cp's OEIS frontend

A191581 Numbers whose sum of their anti-divisors divides the sum of their divisors.

3, 6, 11, 22, 30, 33, 65, 82, 117, 218, 354, 483, 508, 537, 3276, 6430, 21541, 117818, 130356, 753612, 1007328, 2113416, 2379540, 3589646, 7231219, 7346148, 8515767, 13050345, 20199648, 34424166, 44575896, 47245905, 50414595, 104335023, 217728002, 1217532421

Offset: 1

Paolo P. Lava, Jun 07 2011

A161917 is a subsequence of this sequence.

			6-> sum divisors=sigma(6)=12; sum anti-divisors=4; 12/4=3.
30-> sum divisors=sigma(30)=72; sum anti-divisors=4+12+20=36; 72/36=2.

Cf. A000203, A066272, A066417, A161917, A191580.

with(numtheory): P:=proc(i) local a, b, j, k, s, n;
for n from 3 to i do b:=divisors(n); s:=add(b[k],k=1..nops(b));
k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if type(s/a,integer) then print(n); fi; od; end: P(10^6);

f[n_] := Total@ Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[Range[3, 10^3], Mod[DivisorSigma[1, #], f@ #] == 0 &] (* _Michael De Vlieger, Oct 08 2015 *)

{n: A066417(n) | A000203(n)}. - R. J. Mathar, Oct 01 2011

a(21)-a(36) from Donovan Johnson, Jun 24 2012

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A191581 Numbers whose sum of their anti-divisors divides the sum of their divisors.

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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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cp's OEIS Frontend

A191581 Numbers whose sum of their anti-divisors divides the sum of their divisors.

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Author

Keywords

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Maple

Mathematica

Formula

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.

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Author

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Comments

Examples

Crossrefs

Programs

Maple

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.