%I #22 Sep 13 2019 22:39:16
%S 4,9,16,25,36,49,64,81,100,144,169,196,225,256,289,324,361,441,484,
%T 529,576,625,676,729,784,841,900,961,1024,1089,1156,1225,1296,1369,
%U 1444,1681,1764,1849,1936,2025,2116,2209,2304,2500,2601,2704,2809,2916,3025,3136
%N Squares m such that beta(m) = (tau(m) - 3)/2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.
%C As tau(m) = 2 * beta(m) + 3 is odd, the terms of this sequence are squares.
%C There are two classes of terms in this sequence (see examples):
%C 1) Non-Brazilian squares of primes; as tau(p^2) = 3, thus beta(p^2) = (tau(p^2) - 3)/2 = 0, these squares of primes form A326708.
%C 2) Squares of composites which have no Brazilian representation with three digits or more, these integers form A326709.
%C The corresponding square roots are: 2, 3, 4, 5, 6 ,7 ,8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, ...
%C As the number of Brazilian representations of a square m with repdigits of length = 2 is beta'(m) = (tau(m) - 3)/2, we have always beta(m) >= (tau(m) - 3)/2, thus there are no squares m such as beta(m) = (tau(m) - k)/2 with some k >= 5.
%H Bernard Schott, <a href="/A326707/a326707_1.pdf">Abstract - Relations beta = f(tau)</a>
%H Bernard Schott, <a href="/A326707/a326707.pdf">Array - Relations beta = f(tau) for squares</a>
%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>
%e One example for each type:
%e 25 = 5^2, tau(25) = 3 and beta(25) = 0 because 25 is not Brazilian.
%e 196 = 14^2 = 77_27 = 44_48 = 22_97, so beta(196) = 3 with tau(196) = 9 and (9-3)/2 = 3.
%t brazQ[n_, b_] := Length@Union@IntegerDigits[n, b] == 1; beta[n_] := Sum[Boole @ brazQ[n, b], {b, 2, n - 2}]; aQ[n_] := beta[n] == (DivisorSigma[0, n] - 3)/2; Select[Range[56]^2, aQ] (* _Amiram Eldar_, Sep 06 2019 *)
%Y Cf. This sequence (tau(m)-3)/2, A326710 (tau(m)-1)/2.
%Y Subsequences: A326708, A326709.
%Y Subsequence of A000290.
%K nonn,base
%O 1,1
%A _Bernard Schott_, Aug 26 2019