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A387333 a(n) is the least k having exactly n divisors that are not balanced numbers.

%I A387333 #16 Sep 01 2025 09:02:24
%S A387333 1,4,8,16,20,48,40,72,80,120,144,180,320,240,440,432,540,360,792,864,
%T A387333 900,960,1620,720,1080,1920,2808,2016,2340,1440,3168,3024,2160,4032,
%U A387333 5616,2880,5940,8100,3600,6048,3960,5040,6480,10920,7560,14112,11700,7200,8640,11880,13104,13680,7920,10080
%N A387333 a(n) is the least k having exactly n divisors that are not balanced numbers.
%C A387333 a(n) is the least k such that A386591(k) = n.
%H A387333 David A. Corneth, <a href="/A387333/b387333.txt">Table of n, a(n) for n = 0..752</a> (first 251 terms from Robert Israel)
%e A387333 a(3) = 16 because 16 has exactly 3 divisors that are not balanced numbers, namely 4, 8 and 16, and no smaller number works.
%p A387333 g:= proc(n) option remember; numtheory:-sigma(n) mod numtheory:-phi(n) <> 0 end proc:
%p A387333 f:= n -> nops(select(g,numtheory:-divisors(n))):
%p A387333 N:= 60: # for a(0) to a(N)
%p A387333 V:= Array(0..N,-1): count:= 0:
%p A387333 for i from 1 while count < N+1 do
%p A387333   v:= f(i);
%p A387333   if V[v] = -1 then count:= count+1; V[v]:= i; fi
%p A387333 od:
%p A387333 convert(V,list);
%o A387333 (PARI) a(n) = my(k=1); while (sumdiv(k, d, sigma(d)%eulerphi(d) != 0) != n, k++); k; \\ _Michel Marcus_, Aug 26 2025
%Y A387333 Cf. A020492, A386591.
%K A387333 nonn,new
%O A387333 0,2
%A A387333 _Robert Israel_, Aug 26 2025