A326708 -id:A326708 - cp's OEIS frontend

A326707 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.

4, 9, 16, 25, 36, 49, 64, 81, 100, 144, 169, 196, 225, 256, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025, 3136

Offset: 1

Bernard Schott, Aug 26 2019

As tau(m) = 2 * beta(m) + 3 is odd, the terms of this sequence are squares.
There are two classes of terms in this sequence (see examples):
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.
2) Squares of composites which have no Brazilian representation with three digits or more, these integers form A326709.
The corresponding square roots are: 2, 3, 4, 5, 6 ,7 ,8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, ...
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.

			One example for each type:
25 = 5^2, tau(25) = 3 and beta(25) = 0 because 25 is not Brazilian.
196 = 14^2 = 77_27 = 44_48 = 22_97, so beta(196) = 3 with tau(196) = 9 and (9-3)/2 = 3.

Bernard Schott, Abstract - Relations beta = f(tau)
Bernard Schott, Array - Relations beta = f(tau) for squares
Index entries for sequences related to Brazilian numbers

Cf. This sequence (tau(m)-3)/2, A326710 (tau(m)-1)/2.
Subsequences: A326708, A326709.
Subsequence of A000290.

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 *)

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.

Original entry on oeis.org

16, 36, 64, 81, 100, 144, 196, 225, 256, 324, 441, 484, 576, 625, 676, 729, 784, 900, 1024, 1089, 1156, 1225, 1296, 1444, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2916, 3025, 3136, 3249, 3364, 3600, 3844, 3969, 4096, 4225, 4356, 4624, 4761, 4900, 5184, 5476, 5625

Offset: 1

Bernard Schott, Aug 29 2019

This sequence is the second subsequence of A326707: squares of composites which have no Brazilian representation with three digits or more.
As tau(m) = 2 * beta(m) + 3, the number of divisors of these squares of composites m is odd with tau(m) >= 5.
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, ...

			a(1) = 16: tau(16) = 5 and beta(16) = 1 with 16 = 4^2 = 22_7.
a(3) = 64: tau(64) = 7 and beta(64) = 2 with 64 = 8^2 = 44_15 = 22_31.
a(5) = 100: tau(100) = 9 and beta(100) = 3 with 100 = 10^2 = 55_19 = 44_24 = 22_49.

Index entries for sequences related to Brazilian numbers

Subsequence of A000290.
Intersection of A062312 and A326707.
Cf. A326707 = A326708 Union {this sequence} with empty intersection.
Cf. A048691 (number of divisors of n^2).
Cf. A000005 (tau), A220136 (beta).

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 *)

cp's OEIS Frontend

A326707 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.

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