A135776 Numbers having number of divisors equal to number of digits in base 6.

1, 7, 11, 13, 17, 19, 23, 29, 31, 49, 121, 169, 217, 218, 219, 221, 226, 235, 237, 247, 249, 253, 254, 259, 262, 265, 267, 274, 278, 287, 291, 295, 298, 299, 301, 302, 303, 305, 309, 314, 319, 321, 323, 326, 327, 329, 334, 335, 339, 341, 343, 346, 355, 358

Offset: 1

Author

M. F. Hasler, Nov 28 2007

Keywords

• base
• nonn

Comments

Since 6 is not a prime, no element > 1 of the sequence A000400(k)=6^k (having k+1 digits in base 6, but much more divisors) can be a member of this sequence. However, all powers of 7 up to 7^11 are in this sequence, having the same number of digits (in base 6) as the same power of 6 (since 11 = floor(log(7/6)/log(6))) and also that number of divisors (since 7 is prime).

Examples

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 6 as in any other base).
They are followed by the primes (having 2 divisors {1,p}) between 6 and 6^2 - 1 (to have 2 digits in base 6).
Then come the squares of primes (3 divisors) between 6^2 = 100_6 and 6^3 - 1 = 555_6.
These are followed by all semiprimes and cubes of primes (4 divisors) between 6^3 and 6^4 - 1.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Abel Jansma, E_8 Symmetry Structures in the Ising model, Master's thesis, University of Amsterdam, 2018.

Crossrefs

Cf. A135772-A135779, A095862.

Programs

Mathematica

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 6] &] (* G. C. Greubel, Nov 08 2016 *)

PARI

for(d=1,4,for(n=6^(d-1),6^d-1,d==numdiv(n)&print1(n", ")))