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.

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

1,1

Comments

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, ...

Links

Table of n, a(n) for n=1..49.

Index entries for sequences related to Brazilian numbers

Example

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.

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[Select[Range[75], CompositeQ]^2, aQ] (* Amiram Eldar, Sep 06 2019 *)

Crossrefs

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).

Sequence in context: A144548 A161753 A062312 * A331139 A031360 A295016

Adjacent sequences: A326706 A326707 A326708 * A326710 A326711 A326712

Keyword

nonn,base

Author

Bernard Schott, Aug 29 2019

Status

approved