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

Author

M. F. Hasler, Nov 28 2007

Keywords

• base
• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A135772-A135779, A095862.

Programs

Mathematica

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

PARI

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