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A167401 a(n) is the smallest number k such that n*k has twice as many divisors as k.

%I A167401 #19 Mar 17 2022 05:52:12
%S A167401 1,1,2,1,12,1,4,3,20,1,72,1,28,45,8,1,108,1,160,63,44,1,288,5,52,9,
%T A167401 224,1,10800,1,16,99,68,175,864,1,76,117,800,1,21168,1,352,675,92,1,
%U A167401 1152,7,400,153,416,1,648,275,1568,171,116,1,259200
%N A167401 a(n) is the smallest number k such that n*k has twice as many divisors as k.
%C A167401 a(n) is 1 for all prime numbers n.
%C A167401 From _Robert Israel_, Feb 09 2017: (Start)
%C A167401 All prime factors of a(n) divide n.
%C A167401 If n=p^k is a prime power, a(n) = p^(k-1).
%C A167401 If n=p*q with p<q is in A006881, a(n) = p^2*q. (End)
%H A167401 Robert Israel, <a href="/A167401/b167401.txt">Table of n, a(n) for n = 2..10000</a>
%p A167401 A167401 := proc(n) if isprime(n) then 1; else for a from 2 do if numtheory[tau](n*a) = 2*numtheory[tau](a) then return a ; end if; end do ; fi; end: seq(A167401(n),n=2..60) ; # _R. J. Mathar_, Nov 04 2009
%t A167401 tmd[n_]:=Module[{a=1},While[DivisorSigma[0,a*n]!=2DivisorSigma[0,a],a++];a]; Array[tmd,60,2] (* _Harvey P. Dale_, Apr 20 2013 *)
%o A167401 (PARI) a(n) = {my(k=1); while (numdiv(n*k) != 2*numdiv(k), k++); k;} \\ _Michel Marcus_, Feb 10 2017
%Y A167401 Cf. A139315.
%K A167401 nonn,look
%O A167401 2,3
%A A167401 _J. Lowell_, Nov 02 2009
%E A167401 Extended by _Ray Chandler_, Nov 10 2009
%E A167401 Extended beyond a(10) by _R. J. Mathar_, Nov 04 2009