A273992 Numbers whose sum of anti-divisors is equal to the sum of its unitary divisors.
11, 22, 33, 65, 82, 140, 218, 228, 483, 537, 616, 1184, 2889, 6430, 10216, 15849, 21541, 59620, 112590, 117818, 130356, 483153, 3028671, 3589646, 7231219, 8515767, 13050345, 36494625, 44498344, 50414595, 217728002, 459644211, 519061576, 1217532421, 1573368218
Offset: 1
Keywords
Examples
Sum of anti-divisors of 11 is 12. Unitary divisors of 11 are 1, 11 and their sum is 12.
Programs
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Maple
with(numtheory): P:=proc(q) local a,b,c,j,k,n; for n from 1 to q do 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; c:=ifactors(n)[2]; b:=mul(c[j][1]^c[j][2]+1,j=1..nops(c)); if a=b then print(n); fi; od; end: P(10^6);
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Mathematica
Select[Range[5000], Function[n, Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]] == Plus @@ Select[Divisors@ n, GCD[#, n/#] == 1 &]]] (* _Michael De Vlieger, Jun 06 2016, after Robert G. Wilson v at A034448 and Harvey P. Dale at A066272 *)
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PARI
sud(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)); sad(n) = my(k); if(n>1, k=valuation(n, 2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2, 0); isok(n) = sad(n) == sud(n); \\ Michel Marcus, Jun 12 2016
Extensions
a(23)-a(26) from Michel Marcus, Jun 12 2016
a(27)-a(35) from Amiram Eldar, Jul 12 2022