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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..50.

Bernard Schott, Abstract - Relations beta = f(tau)

Bernard Schott, Array - Relations beta = f(tau) for squares

Index entries for sequences related to Brazilian numbers

Example

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.

Mathematica

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 *)

Crossrefs

Cf. This sequence (tau(m)-3)/2, A326710 (tau(m)-1)/2.

Subsequences: A326708, A326709.

Subsequence of A000290.

Sequence in context: A179459 A325148 A325149 * A352618 A030476 A077355

Adjacent sequences: A326704 A326705 A326706 * A326708 A326709 A326710

Keyword

nonn,base

Author

Bernard Schott, Aug 26 2019

Status

approved