A326382 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.

32767, 65535, 67053, 2097151, 4381419, 7174453, 9808617, 13938267, 14348906, 19617234, 21523360, 29425851, 39234468, 43046720, 48686547, 49043085, 58851702, 61035156, 68660319, 71270178, 78468936, 88277553, 98086170, 107894787, 115174101, 117703404, 134217727, 142540356, 175965517

Offset

1,1

Comments

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

There are 2 subsequences which realize a partition of this sequence (see array in link and examples):

1) Non-oblong composites which have exactly three Brazilian representations with three digits or more, they are in A326389.

2) Oblong numbers that have exactly four Brazilian representations with three digits or more. These integers have been found through b-file of Rémy Sigrist in A290869. These oblong integers are a subsequence of A309062.

There are no primes that satisfy this relation.

Links

Table of n, a(n) for n=1..29.

Bernard Schott, Array of relations beta = f(tau)

Index entries for sequences related to Brazilian numbers

Example

One example for each type:
1) The divisors of 32767 are {1, 7, 31, 151, 217, 1057, 4681, 32767} and tau(32767) = 8; also, 32767 = M_15 = R(15)_2 = 77777_8 = (31,31,31)_32 = (151,151)_216 = (31,31)_1056 = 77_4680 so beta(32767) = 6 with beta'(32767) = 3 and beta"(32767)= 3. The relation is beta(32767) = tau(32767)/2 + 2 = 6.
2) 61035156 = 7812 * 7813 is oblong with tau(61035156) = 144. The four Brazilian representations with three digits or more are 61035156 = R(12)_5 = 666666_25 = (31,31,31,31)_125 = (156,156,156)_625, so beta"(61035156) = 4 and beta(61035156) = tau(61035156)/2 + 2 = 74.

Crossrefs

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

Subsequence of A167782, A167783 and A290869.

Cf. A326378 (tau(m)/2 - 2), A326379 (tau(m)/2 - 1), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), this sequence (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).

Cf. A309062, A326389.

Sequence in context: A101744 A013691 A152932 * A326389 A075966 A011565

Adjacent sequences: A326379 A326380 A326381 * A326383 A326384 A326385

Keyword

nonn,base

Author

Bernard Schott, Jul 08 2019

Extensions

Missing a(18) inserted by Bernard Schott, Jul 20 2019

Status

approved