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

1,2

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).

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A135772-A135779, A095862.

Sequence in context: A020603 A283532 A163648 * A067831 A086998 A028416

Adjacent sequences: A135773 A135774 A135775 * A135777 A135778 A135779

Keyword

base,nonn

Author

M. F. Hasler, Nov 28 2007

Status

approved