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

1, 2, 3, 4, 8, 11, 24, 83, 85, 125, 152, 156, 175, 227, 297, 365, 443, 445, 533, 584, 600, 629, 847, 924, 965, 969, 1036, 1091, 1304, 1308, 1458, 1523, 1612, 1685, 1800, 1853, 1960, 2027, 2316, 2340, 2409, 2605, 2716, 2813, 3029, 3251, 3729, 3973, 4108, 4233

Offset: 1

Author

Robert Israel, Jul 20 2014

Keywords

• nonn

Comments

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

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)

Crossrefs

Cf. A000005, A049820, A245197.

Programs

Maple

filter:= proc(n) local t;
  t:= numtheory:-tau(n);
  issqr(n-t)
end proc;
select(filter, [$1..10^4]);

Mathematica

Select[Range[10^4], IntegerQ[Sqrt[# - DivisorSigma[0, #]]]&] (* Jean-François Alcover, Apr 12 2019 *)

PARI

isok(k) = issquare(k - numdiv(k)); \\ Amiram Eldar, Feb 01 2025

Sage

def is_A245388(n):
    a = sloane.A000005
    return is_square(n - a(n))
A245388_list = lambda up_to: filter(is_A245388, (1..up_to))
A245388_list(4333) # Peter Luschny, Jul 20 2014