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A326378 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 A326378 #57 Feb 02 2025 02:27:31
%S A326378 6,12,20,30,56,72,90,110,132,210,240,272,306,380,420,462,506,552,600,
%T A326378 650,702,756,812,870,930,992,1056,1122,1190,1260,1332,1482,1560,1722,
%U A326378 1806,1892,1980,2070,2162,2256,2352,2450,2550,2652,2756,2862,2970,3080,3192,3306,3422,3540,3660,3782
%N A326378 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 A326378 As tau(m) = 2 * (2 + beta(m)), the terms of this sequence are not squares. Indeed, there exists only one family that satisfies this relation and these integers are exactly the oblong numbers that have no Brazilian representation with three digits or more.
%C A326378 There are no integers such as beta(m) = tau(m)/2 - q with q >= 3.
%H A326378 Amiram Eldar, <a href="/A326378/b326378.txt">Table of n, a(n) for n = 1..800</a>
%H A326378 Bernard Schott, <a href="/A326378/a326378_2.pdf">Relation beta = f(tau)</a>.
%H A326378 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>.
%e A326378 1) tau(m) = 4 and beta(m) = 0: m = 6 which is not Brazilian.
%e A326378 2) tau(m) = 6 and beta(m) = 1: m = 12, 20.
%e A326378    12 = 3 * 4 = 22_5, 20 = 4 * 5 = 22_9.
%e A326378 3) tau(m) = 8 and beta(m) = 2: m = 30, 56, 110, 506, 2162, 3422, ...
%e A326378    30 = 5 * 6 = 33_9 = 22_14, 56 = 7 * 8 = 44_13 = 22_27.
%e A326378 4) tau(m) = 10 and beta(m) = 3: m = 272, ...
%e A326378    272 = 16 * 17 = 88_32 = 44_67 = 22_135.
%e A326378 5) tau(m) = 12 and beta(m) = 4: m = 72, 90, 132, 306, 380, 650, 812, 992, ...
%e A326378    72 = 8 * 9 = 66_11 = 44_17 = 33_23 = 22_35.
%o A326378 (PARI) beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
%o A326378 isok(n) = beta(n) == numdiv(n)/2 - 2; \\ _Michel Marcus_, Jul 08 2019
%Y A326378 Cf. A000005 (tau), A220136 (beta).
%Y A326378 Subsequence of A002378 (oblong numbers).
%Y A326378 Cf. A326379 (tau(m)/2 - 1), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), A326382 (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).
%Y A326378 Cf. A326384 (oblongs with tau(m)/2 - 1), A326385 (oblongs with tau(m)/2).
%K A326378 nonn,base
%O A326378 1,1
%A A326378 _Bernard Schott_, Jul 02 2019