cp's OEIS Frontend

A178029 Numbers whose sum of divisors equals the sum of their anti-divisors.

%I A178029 #25 Jun 13 2024 02:40:48
%S A178029 11,22,33,65,82,117,218,483,508,537,6430,21541,117818,3589646,7231219,
%T A178029 8515767,13050345,47245905,50414595,104335023,217728002,1217532421,
%U A178029 1573368218,1875543429,2269058065,11902221245,12196454655,12658724029
%N A178029 Numbers whose sum of divisors equals the sum of their anti-divisors.
%F A178029 {n: A066417(n) = A000203(n)}. - _R. J. Mathar_, May 24 2010
%e A178029 6430 is in the sequence because the sum of divisors is 1+2+5+10+643+1286+3215+6430 = 11592
%e A178029 which equals the sum of anti-divisors 3+4+7+9+11+20+77+167+1169+1429+1837+2572+4287 = 11592.
%e A178029 21541 is in the sequence because the sum of divisors is 1+13+1657+21541 = 23212
%e A178029 and equals the sum of anti-divisors 2+3+9+26+67+643+3314+4787+14361 = 23212.
%p A178029 with(numtheory): P:=proc(q) local j,k; k:=0; j:=q; while j mod 2<>1 do k:=k+1; j:=j/2; od; if sigma(q)=sigma(2*q+1)+sigma(2*q-1)+sigma(q/2^k)*2^(k+1)-6*q-2 then q; fi; end: seq(P(i),i=3..10^5);
%p A178029 # alternative Maple implementation:
%p A178029 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:
%p A178029 A066417 := proc(n) add(d,d=antidivisors(n)) ; end proc:
%p A178029 isA178029 := proc(n) numtheory[sigma](n) = A066417(n) ; end proc:
%p A178029 for n from 1 do if isA178029(n) then printf("%d,\n",n) ; end if; end do:
%p A178029 # _R. J. Mathar_, May 24 2010
%t A178029 antidivisors[n_] := Select[Range[2, n-1], Abs[Mod[n, #] - #/2] < 1&];
%t A178029 For[k = 1, k <= 10^5, k++, If[DivisorSigma[1, k] == Total[antidivisors[k]], Print[k]]] (* _Jean-François Alcover_, Jun 14 2023 *)
%o A178029 (Python)
%o A178029 from sympy import divisors
%o A178029 [n for n in range(1,10**5) if sum([d for d in range(2,n) if (n % d) and (2*n) % d in [d-1,0,1]]) == sum(divisors(n))] # _Chai Wah Wu_, Aug 07 2014
%Y A178029 Cf. A000203, A066272, A066417.
%K A178029 nonn
%O A178029 1,1
%A A178029 _Paolo P. Lava_ and _Giorgio Balzarotti_, May 19 2010
%E A178029 a(13)-a(28) from _Donovan Johnson_, Jun 12 2010