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%I #23 Jun 24 2023 09:23:10
%S 6,12,18,20,24,28,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,
%T 100,102,104,105,108,110,112,114,120,126,132,140,144,150,156,160,162,
%U 168,176,180,192,196,198,200,204,208,210,216,220,224,225,228,234,240,252,260
%N Numbers with more divisors than digits in their binary representation.
%C Not all terms are perfect or abundant, with 105 being the first deficient term.
%C There are no primes in the sequence, and 6 is the only semiprime.
%C By the same comments as those at A175495, this sequence is infinite.
%C This sequence is a subsequence of A175495.
%C 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.
%C The number of terms with m binary digits is Sum_{k>m} A346730(m,k). - _Jon E. Schoenfield_, Jul 31 2021
%e 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.
%t Select[Range[1000], (DivisorSigma[0, #] > Floor[1 + Log2[#]]) &]
%o (PARI) isok(m) = numdiv(m) > #binary(m); \\ _Michel Marcus_, Jul 29 2021
%o (Python)
%o from sympy import divisor_count
%o def ok(n): return divisor_count(n) > n.bit_length()
%o print(list(filter(ok, range(1, 261)))) # _Michael S. Branicky_, Jul 29 2021
%Y Cf. A000005, A070939.
%Y Cf. A135772 (equal number rather than more).
%Y Cf. A175495 (where "binary digits in n" is replaced by "log_2(n)").
%K nonn,base,easy
%O 1,1
%A _Alex Meiburg_, Jul 29 2021