A135778 -id:A135778 - cp's OEIS frontend

A135772 Numbers having equal number of divisors and binary digits.

1, 2, 3, 4, 8, 10, 14, 15, 16, 32, 44, 45, 50, 52, 63, 64, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 256, 441, 484, 512, 567, 592, 656, 688, 752, 848, 891, 944, 976

Offset: 1

M. F. Hasler, Nov 28 2007

			a(1) = 1 since 1 has 1 divisor and 1 binary digit.
a(2), a(3) = 2, 3 since 2 = 10_2 and 3 = 11_2 have 2 divisors and 2 binary digits.
a(4) = 4 = 100_2 is the only number with 3 binary digits having 3 divisors.
8, 10, 14, 15 have 4 binary digits and 4 divisors.

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

Cf. A000005, A070939.

Cf. A135773, A135774, A135775, A135776, A135777, A135778, A135779, A095862.

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 2] &] (* G. C. Greubel, Nov 08 2016 *)

for(d=1,10,for(n=2^(d-1),2^d-1,d==numdiv(n)&print1(n", ")))

from sympy import divisor_count
def ok(n): return divisor_count(n) == n.bit_length()
print(list(filter(ok, range(1, 977)))) # Michael S. Branicky, Jul 29 2021

A135779 Numbers having number of divisors equal to number of digits in base 9.

Original entry on oeis.org

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 121, 169, 289, 361, 529, 731, 734, 737, 745, 746, 749, 753, 755, 758, 763, 766, 767, 771, 778, 779, 781, 785, 789, 791, 793, 794, 799, 802, 803, 807, 813, 815, 817

Offset: 1

M. F. Hasler, Nov 28 2007

Since 9 is not a prime, no element > 1 of the sequence A001019(k)=9^k (having k+1 digits in base 9, but 2k+1 divisors) can be member of this sequence. Also, no power of a prime less than 9 can be in the sequence, since it will always have fewer divisors than digits in base 9. However all powers of 11 up to 11^10 are in this sequence, having the same number of digits (in base 9) than the same power of 9 (since 10 = floor(log(11/9)/log(9))) and also that number of divisors (since 11 is prime).

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 9 as in any other base).
It is followed by the primes (having 2 divisors {1,p}) between 9 and 9^2 - 1 (to have 2 digits in base 9).
Then come the squares of primes (3 divisors) between 9^2 = 100_9 and 9^3 - 1 = 888_9.
These are followed by all semiprimes and cubes of primes (4 divisors) between 9^3 and 9^4 - 1.

G. C. Greubel, Table of n, a(n) for n = 1..1500

Cf. A135772-A135778, A095862.

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 9] &] (* G. C. Greubel, Nov 09 2016 *)

for(d=1,4,for(n=9^(d-1),9^d-1,d==numdiv(n)&print1(n", ")))

cp's OEIS Frontend

A135772 Numbers having equal number of divisors and binary digits.

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