cp's OEIS Frontend

A326379 Numbers m such that beta(m) = tau(m)/2 - 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

%I A326379 #41 Feb 02 2025 02:27:14
%S A326379 2,3,5,8,10,11,14,17,18,19,22,23,24,27,28,29,32,33,34,35,37,38,39,41,
%T A326379 42,44,45,46,47,48,50,51,52,53,54,55,58,59,60,61,65,66,67,68,69,70,71,
%U A326379 74,75,76,77,78,79,82,83,84,87,88,89,92,94,95,96,97,98,99,101,102,103,104,105,106,107,108,109,112,113,115,116
%N A326379 Numbers m such that beta(m) = tau(m)/2 - 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.
%C A326379 As tau(m) = 2 * (1 + beta(m)), the terms of this sequence are not squares. Indeed, there are 3 subsequences which realize a partition of this sequence (see examples):
%C A326379 1) Non-oblong composites which have no Brazilian representation with three digits or more, they form A326386.
%C A326379 2) Oblong numbers that have only one Brazilian representation with three digits or more. These oblong integers are a subsequence of A167782 and form A326384.
%C A326379 3) Non Brazilian primes, then beta(p) = tau(p)/2 - 1 = 0.
%H A326379 Amiram Eldar, <a href="/A326379/b326379.txt">Table of n, a(n) for n = 1..10000</a>
%H A326379 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>.
%e A326379 One example for each type:
%e A326379 10 = 22_4 and tau(10) = 4 with beta(10) = 1.
%e A326379 42 = 6 * 7 = 222_4 = 33_13 = 22_20 and tau(42) = 8 with beta(42) = 3.
%e A326379 17 is no Brazilian prime with tau(17) = 2 and beta(17) = 0.
%o A326379 (PARI) beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
%o A326379 isok(n) = beta(n) == numdiv(n)/2 - 1; \\ _Michel Marcus_, Jul 03 2019
%Y A326379 Cf. A000005 (tau), A220136 (beta).
%Y A326379 Cf. A220627 (subsequence of non Brazilian primes).
%Y A326379 Cf. A326378 (tau(m)/2 - 2), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), A326382 (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).
%K A326379 nonn,base
%O A326379 1,1
%A A326379 _Bernard Schott_, Jul 03 2019