cp's OEIS Frontend

A326709 Squares of composites such that beta(m) = (tau(m) - 3)/2 where beta(m) = A220136(m) is the number of Brazilian representations of m and tau(m) = A000005(m) is the number of divisors of m.

%I A326709 #20 Sep 19 2019 09:40:41
%S A326709 16,36,64,81,100,144,196,225,256,324,441,484,576,625,676,729,784,900,
%T A326709 1024,1089,1156,1225,1296,1444,1764,1936,2025,2116,2304,2500,2601,
%U A326709 2704,2916,3025,3136,3249,3364,3600,3844,3969,4096,4225,4356,4624,4761,4900,5184,5476,5625
%N A326709 Squares of composites such that beta(m) = (tau(m) - 3)/2 where beta(m) = A220136(m) is the number of Brazilian representations of m and tau(m) = A000005(m) is the number of divisors of m.
%C A326709 This sequence is the second subsequence of A326707: squares of composites which have no Brazilian representation with three digits or more.
%C A326709 As tau(m) = 2 * beta(m) + 3, the number of divisors of these squares of composites m is odd with tau(m) >= 5.
%C A326709 The corresponding composites are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 42, ...
%H A326709 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>
%e A326709 a(1) = 16: tau(16) = 5 and beta(16) = 1 with 16 = 4^2 = 22_7.
%e A326709 a(3) = 64: tau(64) = 7 and beta(64) = 2 with 64 = 8^2 = 44_15 = 22_31.
%e A326709 a(5) = 100: tau(100) = 9 and beta(100) = 3 with 100 = 10^2 = 55_19 = 44_24 = 22_49.
%t A326709 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[Select[Range[75], CompositeQ]^2, aQ] (* _Amiram Eldar_, Sep 06 2019 *)
%Y A326709 Subsequence of A000290.
%Y A326709 Intersection of A062312 and A326707.
%Y A326709 Cf. A326707 = A326708 Union {this sequence} with empty intersection.
%Y A326709 Cf. A048691 (number of divisors of n^2).
%Y A326709 Cf. A000005 (tau), A220136 (beta).
%K A326709 nonn,base
%O A326709 1,1
%A A326709 _Bernard Schott_, Aug 29 2019