A175495 -id:A175495 - cp's OEIS frontend

A225735 Numbers n such that n < d(n)^(27/10), where d(n) is the number of divisors of n.

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116

Offset: 1

T. D. Noe, May 14 2013

Alternatively, we could write n^10 < d(n)^27. The last odd term is a(995) = 10395.

T. D. Noe, Table of n, a(n) for n = 1..4282 (complete sequence)

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).

Cf. A225729-A225738.

t = {}; Do[If[n < DivisorSigma[0, n]^(27/10), AppendTo[t, n]], {n, 10^7}]; t

A225736 Numbers n such that n < d(n)^(28/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114

Offset: 1

T. D. Noe, May 14 2013

Alternatively, we could write n^5 < d(n)^14. The last odd term is a(2447) = 45045.

T. D. Noe, Table of n, a(n) for n = 1..9372 (complete sequence)

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).

Cf. A225729-A225738.

t = {}; Do[If[n < DivisorSigma[0, n]^(28/10), AppendTo[t, n]], {n, 10^7}]; t

A352504 Numbers k > 1 such that log(A005179(k))/k is a record low.

Original entry on oeis.org

2, 12, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 45, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 108, 112, 120, 128, 140, 144, 160, 168, 180, 192, 200, 210, 216, 224, 240, 252, 256, 270, 280, 288, 300, 320, 336, 360, 384, 400, 420, 432, 448, 480, 504, 512, 540, 560

Offset: 1

Lucas C. D. Jacobs, Mar 18 2022

nonn

It appears that 15 and 45 are the only odd numbers in the sequence.

It also appears that this is a subsequence of A175495.

Lucas C. D. Jacobs, Table of n, a(n) for n = 1..442

Cf. A005179, A175495.

mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; values = Table[mulpar = mp[n] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 560}]; a = {}; min = 1; Do[ratio = Log[values[[i]]]/i; If[ratio < min, min = ratio; AppendTo[a, i]], {i, 2, Length[values]}]; a (* using code from Vaclav Kotesovec *)

A346695 Numbers with more divisors than digits in their binary representation.

Original entry on oeis.org

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

Offset: 1

Alex Meiburg, Jul 29 2021

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.
The number of terms with m binary digits is Sum_{k>m} A346730(m,k). - Jon E. Schoenfield, Jul 31 2021

			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.

Cf. A000005, A070939.
Cf. A135772 (equal number rather than more).
Cf. A175495 (where "binary digits in n" is replaced by "log_2(n)").

Select[Range[1000], (DivisorSigma[0, #] > Floor[1 + Log2[#]]) &]

isok(m) = numdiv(m) > #binary(m); \\ Michel Marcus, Jul 29 2021

from sympy import divisor_count
def ok(n): return divisor_count(n) > n.bit_length()
print(list(filter(ok, range(1, 261)))) # Michael S. Branicky, Jul 29 2021

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A225735 Numbers n such that n < d(n)^(27/10), where d(n) is the number of divisors of n.

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A225736 Numbers n such that n < d(n)^(28/10), where d(n) is the number of divisors of n.

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A352504 Numbers k > 1 such that log(A005179(k))/k is a record low.

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A346695 Numbers with more divisors than digits in their binary representation.

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