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A245388 Numbers k such that k - tau(k) is a perfect square.

%I A245388 #19 Feb 01 2025 08:42:36
%S A245388 1,2,3,4,8,11,24,83,85,125,152,156,175,227,297,365,443,445,533,584,
%T A245388 600,629,847,924,965,969,1036,1091,1304,1308,1458,1523,1612,1685,1800,
%U A245388 1853,1960,2027,2316,2340,2409,2605,2716,2813,3029,3251,3729,3973,4108,4233
%N A245388 Numbers k such that k - tau(k) is a perfect square.
%C A245388 n - tau(n) = A049820(n) is the number of positive integers < n that do not divide n.
%H A245388 Amiram Eldar, <a href="/A245388/b245388.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Robert Israel)
%e A245388 4 - tau(4) = 4 - 3 = 1^2 so 4 is in the sequence.
%p A245388 filter:= proc(n) local t;
%p A245388   t:= numtheory:-tau(n);
%p A245388   issqr(n-t)
%p A245388 end proc;
%p A245388 select(filter, [$1..10^4]);
%t A245388 Select[Range[10^4], IntegerQ[Sqrt[# - DivisorSigma[0, #]]]&] (* _Jean-François Alcover_, Apr 12 2019 *)
%o A245388 (Sage)
%o A245388 def is_A245388(n):
%o A245388     a = sloane.A000005
%o A245388     return is_square(n - a(n))
%o A245388 A245388_list = lambda up_to: filter(is_A245388, (1..up_to))
%o A245388 A245388_list(4333) # _Peter Luschny_, Jul 20 2014
%o A245388 (PARI) isok(k) = issquare(k - numdiv(k)); \\ _Amiram Eldar_, Feb 01 2025
%Y A245388 Cf. A000005, A049820, A245197.
%K A245388 nonn
%O A245388 1,2
%A A245388 _Robert Israel_, Jul 20 2014