A135778 Numbers having number of divisors equal to number of digits in base 8.

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 121, 169, 289, 361, 514, 515, 517, 519, 526, 527, 533, 535, 537, 538, 542, 543, 545, 551, 553, 554, 559, 562, 565, 566, 573, 579, 581, 583, 586, 589, 591, 597, 611, 614, 622, 623, 626, 629, 633, 634

Offset: 1

Author

M. F. Hasler, Nov 28 2007

Keywords

• base
• nonn

Comments

Since 8 is not a prime, no element > 1 of the sequence A001018(k)=8^k (having k+1 digits in base 8, but much more divisors) can be member of this sequence. Also, no power of a prime less than 8 can be in the sequence, since it will always have fewer divisors than digits in base 8. However all powers of 11 up to 11^6 are in this sequence, having the same number of digits (in base 8) than the same power of 8 (since 6 = floor(log(11/8)/log(8))) and also that number of divisors (since 11 is prime).

Examples

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 8 as in any other base).
They are followed by the primes (having 2 divisors {1,p}) between 8 and 8^2 - 1 (to have 2 digits in base 8).
Then come the squares of primes (3 divisors) between 8^2 = 100_8 and 8^3 - 1 = 777_8.
These are followed by all semiprimes and cubes of primes (4 divisors) between 8^3 and 8^4 - 1.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A135772-A135779, A095862.

Programs

Mathematica

Select[Range[1000],IntegerLength[#,8]==DivisorSigma[0,#]&] (* Harvey P. Dale, Mar 04 2016 *)

PARI

for(d=1,4,for(n=8^(d-1),8^d-1,d==numdiv(n)&print1(n", ")))

Extensions

More terms from Harvey P. Dale, Mar 04 2016