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A346695
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Numbers with more divisors than digits in their binary representation.
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0
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6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 225, 228, 234, 240, 252, 260
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OFFSET
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1,1
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COMMENTS
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Not all terms are perfect or abundant, with 105 being the first deficient term.
There are no primes in the sequence, and 6 is the only semiprime.
By the same comments as those at A175495, this sequence is infinite.
This sequence is a subsequence of A175495.
It is natural to conjecture that this sequence has asymptotic density 0. However, after the first three terms where a(n)/n = 6 -- a function which would increase to infinity if the asymptotic density were zero -- it drops, and it seems to take a long time to get that high again. The first time it gets above 5.0 is at a(30243)=151216. Even as high as a(2188516)=10000000, the density is only ~1/4.57.
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LINKS
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EXAMPLE
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12 has 6 divisors: {1,2,3,4,6,12}. 12 is written in binary as 1100, which has 4 digits. Since 6 > 4, 12 is in the sequence.
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MATHEMATICA
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Select[Range[1000], (DivisorSigma[0, #] > Floor[1 + Log2[#]]) &]
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PROG
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(PARI) isok(m) = numdiv(m) > #binary(m); \\ Michel Marcus, Jul 29 2021
(Python)
from sympy import divisor_count
def ok(n): return divisor_count(n) > n.bit_length()
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CROSSREFS
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Cf. A135772 (equal number rather than more).
Cf. A175495 (where "binary digits in n" is replaced by "log_2(n)").
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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