A326388 - OEIS

%I #18 Jul 20 2019 14:19:17

%S 63,255,273,364,511,546,728,777,931,1023,1365,1464,2730,3280,3549,

%T 3783,4557,6560,7566,7812,9114,9331,9841,10507,11349,11718,13671,

%U 14043,14763,15132,15624,16383,18291,18662,18915,19608,19682,21845,22351,22698

%N 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 As tau(m) = 2 * (beta(m) - 1), the terms of this sequence are not squares.

%C The number of Brazilian representations of a non-oblong number m with repdigits of length = 2 is beta'(n) = tau(n)/2 - 1.

%C This sequence is the first subsequence of A326381: non-oblong composites which have exactly two Brazilian representations with three digits or more.

%C Some Mersenne numbers belong to this sequence: M_6, M_8, M_9, M_10, M_14, ...

%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>

%e tau(m) = 4 and beta(m) = 3 for m = 511 with 511 = 111111111_2 = 777_8 = 77_72,

%e tau(m) = 6 and beta(m) = 4 for m = 63 with 63 = 111111_2 = 333_4 = 77_8 = 33_20,

%e tau(m) = 8 and beta(m) = 5 for m = 255 with 255 = 11111111_2 = 3333_4 = (15,15)_16 = 55_50 = 33_84,

%e tau(m) = 12 and beta(m) = 7 for m = 364 with 364 = 111111_3 = 4444_9 = (14,14)_25 = (13,13)_27 = 77_51 = 44_90 = 22_181.

%o (PARI) isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378

%o beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136

%o isok(m) = !isprime(m) && !isoblong(m) && (beta(m) == numdiv(m)/2 + 1); \\ _Michel Marcus_, Jul 15 2019

%Y Cf. A000005 (tau), A220136 (beta).

%Y Subsequence of A167782, A167783, A308874 and A326381.

%Y Cf. A326386 (non-oblongs with tau(m)/2 - 1), A326387 (non-oblongs with tau(m)/2), A326389 (non-oblongs with tau(m)/2 + 2).

%K nonn,base

%O 1,1

%A _Bernard Schott_, Jul 13 2019