A326707 -id:A326707 - cp's OEIS frontend

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

Bernard Schott, Aug 26 2019

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, ...

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

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

Cf. A190300.
Subsequence of A000290 and of A220570 and of A190300.
Intersection of A001248 and A326707.

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 *)

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 *)

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.

Original entry on oeis.org

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

Bernard Schott, Sep 14 2019

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.

			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.

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

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

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

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

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A326708 Non-Brazilian squares of primes.

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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.

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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.

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