A326708 Non-Brazilian squares of primes.

4, 9, 25, 49, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761

Offset: 1

Author

Bernard Schott, Aug 26 2019

Keywords

• nonn
• base

Comments

This sequence is a subsequence of A326707.

For these terms, we have the relations beta'(p^2) = beta"(p^2) = beta(p^2) = (tau(p^2) - 3)/2 = 0.

This sequence = A001248 \ {121} because 121 is the only known square of a prime that is Brazilian (Wikipédia link); 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), hence 121 = 11^2 = (3^5 -1)/2 = 11111_3.

The corresponding square roots are: 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, ...

Examples

			49 = 7^2 is not Brazilian, so beta(49) = 0 with tau(49) = 3.
		

Links

• Bernard Schott, Array relations beta = f(tau) for squares
• Wikipédia, 121 (nombre) (in French)
• Wikipédia, Nombre brésilien (in French)
• Index entries for sequences related to Brazilian numbers

Crossrefs

Cf. A190300.

Subsequence of A000290 and of A220570 and of A190300.

Intersection of A001248 and A326707.

Programs

Mathematica

brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Select[Range[2, 1000], PrimeQ[#^(1/2)]&& brazBases[#] == {} &] (* Metin Sariyar, Sep 05 2019 *)