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A178029 Numbers whose sum of divisors equals the sum of their anti-divisors. 5
11, 22, 33, 65, 82, 117, 218, 483, 508, 537, 6430, 21541, 117818, 3589646, 7231219, 8515767, 13050345, 47245905, 50414595, 104335023, 217728002, 1217532421, 1573368218, 1875543429, 2269058065, 11902221245, 12196454655, 12658724029 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..28.

FORMULA

{n: A066417(n) = A000203(n)}. [R. J. Mathar, May 24 2010]

EXAMPLE

6430 is in the sequence because the sum of divisors is 1+2+5+10+643+1286+3215+6430 = 11592

which equals the sum of anti-divisors 3+4+7+9+11+20+77+167+1169+1429+1837+2572+4287 = 11592.

21541 is in the sequence because the sum of divisors is 1+13+1657+21541 = 23212

and equals the sum of anti-divisors 2+3+9+26+67+643+3314+4787+14361 = 23212.

MAPLE

with(numtheory): P:=proc(i) local a, b, d, k, n; for n from 3 by 1 to i d d:=0; b:=op(divisors(n)); a:=tau(n); for k from 1 by 1 to a do d:=d+b[k]; od; k:=2; a:=0; while k<n do if (k mod 2)=0 then if (n mod k)>0 and (2*n mod k)=0 then a:=a+k; fi; else if (n mod k)>0 and (((2*n-1) mod k)=0 or ((2*n+1) mod k)=0) then a:=a+k; fi; fi; k:=k+1; od; if d=a then print(n); fi; od; end: P(105000);

# alternative Maple implementation R. J. Mathar, May 24 2010:

antidivisors := proc(n) local a, k; a := {} ; for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a := a union {k} ; end if; end do: a ; end proc:

A066417 := proc(n) add(d, d=antidivisors(n)) ; end proc:

isA178029 := proc(n) numtheory[sigma](n) = A066417(n) ; end proc:

for n from 1 do if isA178029(n) then printf("%d, \n", n) ; end if; end do:

PROG

(Python)

from sympy import divisors

[n for n in range(1, 10**5) if sum([d for d in xrange(2, n) if (n % d) and (2*n) % d in [d-1, 0, 1]]) == sum(divisors(n))] # Chai Wah Wu, Aug 07 2014

CROSSREFS

Cf. A066272

Sequence in context: A070022 A004940 A065816 * A273992 A283927 A239018

Adjacent sequences:  A178026 A178027 A178028 * A178030 A178031 A178032

KEYWORD

nonn

AUTHOR

Paolo P. Lava and Giorgio Balzarotti, May 19 2010

EXTENSIONS

Formula and another Maple program from R. J. Mathar, May 24 2010

a(13)-a(28) from Donovan Johnson, Jun 12 2010

STATUS

approved

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Last modified August 25 09:48 EDT 2019. Contains 326324 sequences. (Running on oeis4.)