A326388 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.

63, 255, 273, 364, 511, 546, 728, 777, 931, 1023, 1365, 1464, 2730, 3280, 3549, 3783, 4557, 6560, 7566, 7812, 9114, 9331, 9841, 10507, 11349, 11718, 13671, 14043, 14763, 15132, 15624, 16383, 18291, 18662, 18915, 19608, 19682, 21845, 22351, 22698

Offset

1,1

Comments

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

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

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

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

Links

Table of n, a(n) for n=1..40.

Index entries for sequences related to Brazilian numbers

Example

tau(m) = 4 and beta(m) = 3 for m = 511 with 511 = 111111111_2 = 777_8 = 77_72,
tau(m) = 6 and beta(m) = 4 for m = 63 with 63 = 111111_2 = 333_4 = 77_8 = 33_20,
tau(m) = 8 and beta(m) = 5 for m = 255 with 255 = 11111111_2 = 3333_4 = (15,15)_16 = 55_50 = 33_84,
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.

Prog

(PARI) isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
isok(m) = !isprime(m) && !isoblong(m) && (beta(m) == numdiv(m)/2 + 1); \\ Michel Marcus, Jul 15 2019

Crossrefs

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

Subsequence of A167782, A167783, A308874 and A326381.

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

Sequence in context: A184449 A158676 A157948 * A158684 A063398 A359564

Adjacent sequences: A326385 A326386 A326387 * A326389 A326390 A326391

Keyword

nonn,base

Author

Bernard Schott, Jul 13 2019

Status

approved