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Numbers k such that the number of divisors of k does not divide the sum of divisors of k.
15

%I #41 Apr 25 2024 09:12:06

%S 2,4,8,9,10,12,16,18,24,25,26,28,32,34,36,40,48,50,52,58,63,64,72,74,

%T 75,76,80,81,82,84,88,90,98,100,104,106,108,112,117,120,121,122,124,

%U 128,130,136,144,146,148,152,156,160,162,170,171,172,175

%N Numbers k such that the number of divisors of k does not divide the sum of divisors of k.

%D József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 119, section III.51.

%H Reinhard Zumkeller, <a href="/A049642/b049642.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_number">Arithmetic number</a>.

%F A054025(a(n)) > 0. - _Reinhard Zumkeller_, Jan 06 2012

%F A245656(a(n)) = 0. - _Reinhard Zumkeller_, Jul 28 2014

%p isA049642 := proc(n)

%p if modp(numtheory[sigma](n),numtheory[tau](n)) = 0 then

%p false;

%p else

%p true;

%p end if;

%p end proc:

%p A049642 := proc(n)

%p option remember;

%p if n = 1 then

%p 2;

%p else

%p for a from procname(n-1)+1 do

%p if isA049642(a) then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc: # _R. J. Mathar_, Oct 26 2015

%t Select[Range[175], Mod[DivisorSigma[1, #], DivisorSigma[0, #]] > 0 &] (* _Jayanta Basu_, Mar 28 2013 *)

%o (Haskell)

%o a049642 n = a049642_list !! (n-1)

%o a049642_list = filter ((== 0) . a245656) [1..]

%o -- _Reinhard Zumkeller_, Jan 06 2012

%o (GAP) a:=Filtered([1..180],n->Sigma(n) mod Tau(n)>0);; Print(a); # _Muniru A Asiru_, Jan 25 2019

%o (PARI) is(n) = {my(f = factor(n)); sigma(f) % numdiv(f) > 0;} \\ _Amiram Eldar_, Apr 25 2024

%Y Complement of A003601.

%Y Cf. A000005, A000203.

%Y Cf. A054025, A245656.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_