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A326382 Numbers m such that beta(m) = tau(m)/2 + 2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

%I A326382 #35 Aug 02 2019 14:01:46
%S A326382 32767,65535,67053,2097151,4381419,7174453,9808617,13938267,14348906,
%T A326382 19617234,21523360,29425851,39234468,43046720,48686547,49043085,
%U A326382 58851702,61035156,68660319,71270178,78468936,88277553,98086170,107894787,115174101,117703404,134217727,142540356,175965517
%N A326382 Numbers m such that beta(m) = tau(m)/2 + 2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.
%C A326382 As tau(m) = 2 * (beta(m) - 2) , the terms of this sequence are not squares.
%C A326382 There are 2 subsequences which realize a partition of this sequence (see array in link and examples):
%C A326382 1) Non-oblong composites which have exactly three Brazilian representations with three digits or more, they are in A326389.
%C A326382 2) Oblong numbers that have exactly four Brazilian representations with three digits or more. These integers have been found through b-file of _Rémy Sigrist_ in A290869. These oblong integers are a subsequence of A309062.
%C A326382 There are no primes that satisfy this relation.
%H A326382 Bernard Schott, <a href="/A326382/a326382.pdf">Array of relations beta = f(tau)</a>
%H A326382 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>
%e A326382 One example for each type:
%e A326382 1) The divisors of 32767 are {1, 7, 31, 151, 217, 1057, 4681, 32767} and tau(32767) = 8; also, 32767 = M_15 = R(15)_2 = 77777_8 = (31,31,31)_32 = (151,151)_216 = (31,31)_1056 = 77_4680 so beta(32767) = 6 with beta'(32767) = 3 and beta"(32767)= 3. The relation is beta(32767) = tau(32767)/2 + 2 = 6.
%e A326382 2) 61035156 = 7812 * 7813 is oblong with tau(61035156) = 144. The four Brazilian representations with three digits or more are 61035156 = R(12)_5 = 666666_25 = (31,31,31,31)_125 = (156,156,156)_625, so beta"(61035156) = 4 and beta(61035156) = tau(61035156)/2 + 2 = 74.
%Y A326382 Cf. A000005 (tau), A220136 (beta).
%Y A326382 Subsequence of A167782, A167783 and A290869.
%Y A326382 Cf. A326378 (tau(m)/2 - 2), A326379 (tau(m)/2 - 1), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), this sequence (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).
%Y A326382 Cf. A309062, A326389.
%K A326382 nonn,base
%O A326382 1,1
%A A326382 _Bernard Schott_, Jul 08 2019
%E A326382 Missing a(18) inserted by _Bernard Schott_, Jul 20 2019