A326389 - OEIS

%I #31 Jun 30 2024 12:30:37

%S 32767,65535,67053,2097151,4381419,7174453,9808617,13938267,14348906,

%T 19617234,21523360,29425851,39234468,43046720,48686547,49043085,

%U 58851702,68660319,71270178,78468936,88277553,98086170,107894787,115174101,117703404,134217727,142540356

%N Non-oblong numbers that are repdigits with length > 2 in exactly three bases.

%C The number of Brazilian representations of a non-oblong number m with repdigits of length = 2 is beta'(m) = tau(m)/2 - 1. So, as here beta"(m) = 3, beta(m) = tau(m)/2 + 2 where beta(m) is the number of Brazilian representations of m. So, this sequence is the first subsequence of A326382.

%C As tau(m) = 2 * (beta(m) - 2) is even, the terms of this sequence are not squares.

%C Some Mersenne numbers belong to this sequence: M_15 = a(1), M_16 = a(2), M_21 = a(4), M_27 = a(26), ...

%H Bernard Schott, <a href="/A326389/a326389.pdf">Array of relations beta = f(tau)</a>

%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>

%e tau(m) = 8 and beta(m) = 6 for m = 32767 with 32767 = R(15)_2 = 77777_8 = (31,31,31)_32.

%e tau(m) = 12 and beta(m) = 8 for m = 2097151 with 2097151 = R(21)_2 = 7777777_8 = (127,127,127)_128.

%e tau(m) = 16 and beta(m) = 10 with m = 67053 = (31,31,31)_46 = (21,21,21)_56 = 333_149.

%o (PARI) isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378

%o beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136

%o isok(m) = !isprime(m) && !isoblong(m) && (beta(m) == numdiv(m)/2 + 2); \\ _Jinyuan Wang_, Aug 02 2019

%Y Cf. A000005 (tau), A220136 (beta).

%Y Subsequence of A167782, A167783, A290869, A308874 and A326382.

%Y Cf. A326386 (non-oblongs with tau(m)/2 - 1), A326387 (non-oblongs with tau(m)/2), A326388 (non-oblongs with tau(m)/2 + 1), this sequence (non-oblongs with tau(m)/2 + 2), A326705 (non-oblongs with tau(m)/2 + k, k >=3).

%K nonn,base

%O 1,1

%A _Bernard Schott_, Jul 20 2019