cp's OEIS Frontend

A326705 Non-oblong numbers that are repdigits with length > 2 in more than three bases.

%I A326705 #30 Aug 03 2019 21:41:44
%S A326705 4095,262143,265720,531440,1048575,5592405,11184810,16777215,
%T A326705 122070312,183105468,193710244,244140624,268435455,387420488,435356467
%N A326705 Non-oblong numbers that are repdigits with length > 2 in more than three bases.
%C A326705 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) = r with r >= 4, beta(m) = tau(m)/2 + k with k >= 3 where beta(m) is the number of Brazilian representations of m.
%C A326705 As tau(m) = 2 * (beta(m) - k) is even, the terms of this sequence are not squares.
%C A326705 The terms which have exactly four Brazilian representations with three digits or more form the first subsequence of A326383. Indeed, for the given terms, the number of bases is 4, except for a(8) and a(15) where this number of bases is respectively 5 and 6 (see examples).
%C A326705 Some Mersenne numbers belong to this sequence: M_12 = a(1), M_18 = a(2), M_20 = a(5), M_24 = a(8), M_28 = a(13), M_32, ...
%H A326705 Bernard Schott, <a href="/A326705/a326705.pdf">Array for the relations beta = f(tau)</a>
%H A326705 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>
%e A326705 If beta"(m)is the number of Brazilian representations with three digits or more of the integer m, then:
%e A326705 1) With beta"(m) = 4; tau(4095) = 24 and 4095 has exactly four Brazilian representations with three digits or more: [R(12)]_2 = 333333_4 = 7777_4 = (15,15,15)_16 and 11 representations with 2 digits, so beta(4095) = 15 and k = 3.
%e A326705 2) With beta"(m) = 5; tau(435356467) = 64 and 435356467 has exactly five Brazilian representations with three digits or more: R(12)_6 = 777777_36 = (43,43,43)_216 = (259,259,259)_1296 = (31,31,31)_3747 and has 31 representations with 2 digits, so beta(435356467) = 36 and k = 4.
%e A326705 3) With beta"(m)=6; tau(16777215)= 96 and 16777215 has exactly six Brazilian representations with three digits or more: [R(24)]_2 = 333333333333_4 = 7777777_8 = (15,15,15,15,15,15)_16 = (63,63,63,63)_64 = (255,255,255)_256 and 47 representations with 2 digits, so beta(16777215) = 53 and k = 5.
%Y A326705 Cf. A000005 (tau), A220136 (beta).
%Y A326705 Subsequence of A167782, A167783, A290869 and A308874.
%Y A326705 Cf. A326386 (non-oblongs with tau(m)/2 - 1), A326387 (non-oblongs with tau(m)/2), A326388 (non-oblongs with tau(m)/2 + 1), A326389 (non-oblongs with tau(m)/2 + 2), this sequence (non-oblongs with tau(m/2) + k, k >= 3).
%K A326705 nonn,base,more
%O A326705 1,1
%A A326705 _Bernard Schott_, Jul 21 2019