A326710 Squares m such that beta(m) = (tau(m) - 1)/2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

1, 121, 400, 1521, 1600, 2401, 6084, 17689, 61009, 244036, 294849, 1179396, 1483524, 2653641, 2725801, 2989441, 4717584, 5239521, 7371225, 9591409, 10614564, 11957764, 14447601, 17397241, 18870336, 20277009, 20958084, 23882769, 26904969, 29484900, 38365636, 38825361, 47155689

Offset

1,2

Comments

As tau(m) = 2 * beta(m) + 1 is odd, the terms of this sequence are squares.

There are 3 classes of terms in this sequence (see examples):

1) The singleton {1} with 1^2 = 1.

2) The singleton {121}. Indeed, 121 is the only known square of prime that is Brazilian because 121 is a solution y^q of the Nagell-Ljunggren equation y^q = (b^m-1)/(b-1) with y = 11, q =2, b = 3, m = 5 (see A208242).

3) Squares of composites which have one Brazilian representation with three digits or more. These integers form A326711. We don't know if there exist squares of composites which have two or more Brazilian representations with three digits or more, consequently, there is no sequence with beta(m) = (tau(m) + k)/2, with k odd >= 1.

Links

Table of n, a(n) for n=1..33.

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

Index entries for sequences related to Brazilian numbers

Formula

a(n+1) = (A158235(n))^2 for n >= 1.

Example

One example for each type:
1) 1 is not Brazilian, tau(1) = 1 and beta(1) = (tau(1) - 1)/2 = 0.
2) 121 = 11^2 = 11111_3, tau(121) = 3 and beta(121) = (tau(121) - 1)/2 = 1.
3) 1521 = 39^2 = 333_22 = (13,13)_116 = 99_168 = 33_506. The divisors of 1521 are {1, 3, 9, 13, 39, 117, 169, 507, 1521} so tau(1521) = 9 and beta(1521) = (tau(1521) - 1)/2 = 4.

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] - 1)/2; Select[Range[6867]^2, aQ] (* Amiram Eldar, Sep 14 2019 *)

Crossrefs

Cf. A326707 (tau(m)-3)/2, this sequence (tau(m)-1)/2.

Subsequence of A000290.

Sequence in context: A253328 A257035 A128608 * A144719 A222551 A017282

Adjacent sequences: A326707 A326708 A326709 * A326711 A326712 A326713

Keyword

nonn,base

Author

Bernard Schott, Sep 14 2019

Status

approved