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n such that n - tau(n) is a perfect square.
3

%I #11 Apr 12 2019 09:20:57

%S 1,2,3,4,8,11,24,83,85,125,152,156,175,227,297,365,443,445,533,584,

%T 600,629,847,924,965,969,1036,1091,1304,1308,1458,1523,1612,1685,1800,

%U 1853,1960,2027,2316,2340,2409,2605,2716,2813,3029,3251,3729,3973,4108,4233

%N n such that n - tau(n) is a perfect square.

%C n - tau(n) = A049820(n) is the number of positive integers < n that do not divide n.

%H Robert Israel, <a href="/A245388/b245388.txt">Table of n, a(n) for n = 1..1000</a>

%e 4 - tau(4) = 4 - 3 = 1^2 so 4 is in the sequence.

%p filter:= proc(n) local t;

%p t:= numtheory:-tau(n);

%p issqr(n-t)

%p end proc;

%p select(filter, [$1..10^4]);

%t Select[Range[10^4], IntegerQ[Sqrt[# - DivisorSigma[0, #]]]&] (* _Jean-François Alcover_, Apr 12 2019 *)

%o (Sage)

%o def is_A245388(n):

%o a = sloane.A000005

%o return is_square(n - a(n))

%o A245388_list = lambda up_to: filter(is_A245388, (1..up_to))

%o A245388_list(4333) # _Peter Luschny_, Jul 20 2014

%Y Cf. A000005, A049820, A245197

%K nonn

%O 1,2

%A _Robert Israel_, Jul 20 2014