cp's OEIS Frontend

A326386 Non-oblong composites 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 A326386 #32 Feb 02 2025 02:26:02
%S A326386 8,10,14,18,22,24,27,28,32,33,34,35,38,39,44,45,46,48,50,51,52,54,55,
%T A326386 58,60,65,66,68,69,70,74,75,76,77,78,82,84,87,88,92,94,95,96,98,99,
%U A326386 102,104,105,106,108,112,115,116,117,118,119,120,122,123,125,126,128,130,134,135,136
%N A326386 Non-oblong composites 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 A326386 As tau(m) = 2 * (1 + beta(m)), the terms of this sequence are not squares.
%C A326386 The number of Brazilian representations of a non-oblong number m with repdigits of length = 2 is beta'(n) = tau(n)/2 - 1.
%C A326386 This sequence is the first subsequence of A326379: non-oblong composites which have no Brazilian representation with three digits or more.
%H A326386 Amiram Eldar, <a href="/A326386/b326386.txt">Table of n, a(n) for n = 1..10000</a>
%H A326386 Bernard Schott, <a href="/A326386/a326386_1.pdf">beta = f(tau) - Abstract</a>.
%H A326386 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>.
%e A326386 tau(m) = 4 and beta(m)=1 for m = 8, 10, 14, 22, 27, 33, 34, 35, 38, ... 8 = 22_3,
%e A326386 tau(m) = 6 and beta(m)=2 for m = 18, 28, 32, 44, 45, 50, ... 18 = 33_5 = 22_8,
%e A326386 tau(m) = 8 and beta(m)=3 for m = 24, 54, 66, 70, ... 24 = 44_5 = 33_7 = 22_11,
%e A326386 tau(m) = 10 and beta(m) = 4: 48, 112, ... 48 = 66_7 = 44_11 = 33_15 = 22_23.
%o A326386 (PARI) isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
%o A326386 beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
%o A326386 isok(m) = !isprime(m) && !isoblong(m) && (beta(m) == numdiv(m)/2 - 1); \\ _Michel Marcus_, Jul 15 2019
%Y A326386 Cf. A000005 (tau), A220136 (beta).
%Y A326386 Subsequence of A308874 and of A326379.
%Y A326386 Cf. A326387 (non-oblongs with tau(m)/2), A326388 (non-oblongs with tau(m)/2 + 1), A326389 (non-oblongs with tau(m)/2 + 2).
%K A326386 nonn,base
%O A326386 1,1
%A A326386 _Bernard Schott_, Jul 12 2019