A135774 Numbers having number of divisors equal to number of digits in base 4.

1, 5, 7, 11, 13, 25, 49, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218

Offset

1,2

Comments

Since 4 is not a prime, no element > 1 of the sequence A000302(k)=4^k (having k+1 digits in base 4 but 2k+1 divisors) can be member of this sequence. However all powers of 5 up to 5^6 are in this sequence, having the same number of digits (in base 4) than the same power of 4 (since (5/4)^6 < 4 < (5/4)^7) and also that number of divisors.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Example

a(1) = 1 since 1 has 1 divisor and 1 digit (in base 4 as in any other base).
a(2)..a(5) = 5, 7, 11, 13 are the primes (to have 2 divisors {1,p}) between 4 and 4^2 - 1 (to have 2 digits in base 4).
a(6), a(7) = 25, 49 are the squares of primes (3 divisors) between 4^2 = 100[4] and 4^3 - 1 = 333_4.
They are followed by all semiprimes and cubes of primes (4 divisors) between 4^3 and 4^4 - 1.

Mathematica

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 4] &] (* G. C. Greubel, Nov 08 2016 *)

Prog

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

Crossrefs

Cf. A135772-A135779, A095862.

Sequence in context: A263467 A200143 A265780 * A180946 A265006 A135930

Adjacent sequences: A135771 A135772 A135773 * A135775 A135776 A135777

Keyword

base,nonn

Author

M. F. Hasler, Nov 28 2007

Status

approved