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
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
KEYWORD
nonn,base
AUTHOR
Bernard Schott, Aug 26 2019
STATUS
approved