A336591 -id:A336591 - cp's OEIS frontend

A036537 Numbers whose number of divisors is a power of 2.

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Labos Elemer

nonn

Primes and A030513(d(x)=4) are subsets; d(16k+4) and d(16k+12) have the form 3Q, so x=16k+4 or 16k-4 numbers are missing.
A number m is a term if and only if all its divisors are infinitary, or A000005(m) = A037445(m). - Vladimir Shevelev, Feb 23 2017
All exponents in the prime number factorization of a(n) have the form 2^k-1, k >= 1. So it is an S-exponential sequence (see Shevelev link) with S={2^k-1}. Using Theorem 1, we obtain that a(n) ~ C*n, where C = Product((1-1/p)*(1 + Sum_{i>=1} 1/p^(2^i-1))). - Vladimir Shevelev Feb 27 2017
This constant is C = 0.687827... . - Peter J. C. Moses, Feb 27 2017
From Peter Munn, Jun 18 2022: (Start)
1 and numbers j*m^2, j squarefree, m >= 1, such that all prime divisors of m divide j, and m is in the sequence.
Equivalently, the nonempty set of numbers whose squarefree part (A007913) and squarefree kernel (A007947) are equal, and whose square part's square root (A000188) is in the set.
(End)

			383, 384, 385, 386 have 2, 16, 8, 4 divisors, respectively, so they are consecutive terms of this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithm., 175(2016), 385-395.
Eric Weisstein's World of Mathematics, Square Part, Squarefree Part

A005117, A030513, A058891, A175496, A336591 are subsequences.
Complement of A162643; subsequence of A002035. - Reinhard Zumkeller, Jul 08 2009
Subsequence of A162644, A337533.
Cf. A000005, A000188, A007913, A007947, A036538, A352780.
The closure of the squarefree numbers under application of A355038(.) and lcm.

a036537 n = a036537_list !! (n-1)
a036537_list = filter ((== 1) . a209229 . a000005) [1..]
-- Reinhard Zumkeller, Nov 15 2012

bi[ x_ ] := 1-Sign[ N[ Log[ 2, x ], 5 ]-Floor[ N[ Log[ 2, x ], 5 ] ] ]; ld[ x_ ] := Length[ Divisors[ x ] ]; Flatten[ Position[ Table[ bi[ ld[ x ] ], {x, 1, m} ], 1 ] ]
Select[Range[110],IntegerQ[Log[2,DivisorSigma[0,#]]]&] (* Harvey P. Dale, Nov 20 2016 *)

is(n)=n=numdiv(n);n>>valuation(n,2)==1 \\ Charles R Greathouse IV, Mar 27 2013

isok(m) = issquarefree(m) || (omega(m) == omega(core(m)) && isok(core(m,1)[2])); \\ Peter Munn, Jun 18 2022

from itertools import count, islice
from sympy import factorint
def A036537_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:not((k:=m+1)&-k)^k,factorint(n).values())),count(max(startvalue,1)))
A036537_list = list(islice(A036537_gen(),30)) # Chai Wah Wu, Jan 04 2023

A209229(A000005(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2012
a(n) << n. - Charles R Greathouse IV, Feb 25 2017
m is in the sequence iff for k >= 0, A352780(m, k+1) | A352780(m, k)^2. - Peter Munn, Jun 18 2022

A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A013929 and first differs from it at n = 27: A013929(27) = 72 = 2^3 * 3^2 is not a term of this sequence.
Numbers whose prime factorization has one distinct exponent that does not equal 1.
Numbers that are a product of a squarefree number (A005117) and a power of a different squarefree number that is not squarefree.
The asymptotic density of this sequence is Sum_{k>=2} (d(k)-1)/zeta(2) = 0.36113984820338109927..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

			12 = 2^2 * 3 is a term because its powerful part, 4 = 2^2, is a power of a squarefree number, 2, that is larger than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, Vol. 12, No. 1 (1962), pp. 77-84.
Index entries for sequences computed from exponents in factorization of n.

Cf. A005117, A057521.
Subsequence of A013929.
Subsequences: A067259, A072777, A190641, A336591.

q[n_] := Count[Union[FactorInteger[n][[;; , 2]]], _?(# > 1 &)] == 1; Select[Range[200], q]

is(k) = {my(e = select(x -> (x > 1), Set(factor(k)[,2]))); #e == 1;}

A380922 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(3*s)).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4, 6, 2, 8, 4, 4, 4, 4, 4, 6, 2, 4, 4, 4

Offset: 1

Vaclav Kotesovec, Apr 22 2025

First differs from A061389 at n = 32.
First differs from A322483 at n = 32.
First differs from A372380 at n = 128 (next differences are at n=128*k, n=2187*k, ...).
The number of divisors of n that are both biquadratefree (A046100) and exponentially odd (A268335), i.e., in A336591. - Amiram Eldar, Apr 22 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A073184, A365498, A383292.
Cf. A322483, A372380, A061389.
Cf. A046100, A268335, A336591.

f[p_, e_] := If[e < 3, 2, 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 22 2025 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X + X^3))[n], ", "))

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(4*s)).
Dirichlet g.f.: zeta(s)^2 * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.684286924186862318141968725791218083472312736723163777284618226290055...,
f'(1) = f(1) * Sum_{p prime} (2*p^2 - 3*p + 4) * log(p) / ((p-1) * (p^3 + p^2 + 1)) = f(1) * 0.85825768698295295413525347933038488513032293516964600096226328323449...
and gamma is the Euler-Mascheroni constant A001620.
Multiplicative with a(p^e) = 2 if e <= 2 and 3 otherwise. - Amiram Eldar, Apr 22 2025

A375074 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no higher exponents.

Original entry on oeis.org

72, 108, 200, 360, 392, 500, 504, 540, 600, 675, 756, 792, 936, 968, 1125, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1500, 1656, 1800, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2312, 2484, 2520, 2600, 2646, 2664, 2700, 2888, 2904, 2952, 3087

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers whose powerful part (A057521) is a term of A375073.

The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) + 1/zeta(2) - zeta(6)/(zeta(2) * zeta(3)) * c = A215267 - A088453 + A059956 - A068468 * c = 0.0156712080080470088619..., where c = Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5).

Equals A046100 \ (A004709 UNION A336591).
Disjoint union of A375073 and A375075.
Cf. A051903, A057521.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A215267.

Select[Range[3000], Union[Select[FactorInteger[#][[;; , 2]], # > 1 &]] == {2, 3} &]

is(k) = Set(select(x -> x > 1, factor(k)[,2])) == [2, 3];

A051903(a(n)) = 3.

A375076 Numbers whose prime factorization exponents include at least one 1, at least one 3 and no other exponents.

Original entry on oeis.org

24, 40, 54, 56, 88, 104, 120, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696, 702, 712, 728, 744, 750

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A360793 at n = 79: a(79) = 1080 = 2^3 * 3^3 * 5 is not a term of A360793.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 3}.
The asymptotic density of this sequence is ((zeta(6)/zeta(3)) * Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5) - 1)/zeta(2) = 0.076359822332835689478... .

Equals A336591 \ (A005117 UNION A062838).
Subsequences: A065036, A360793.
Cf. A002117, A013661, A013664, A136568.

Select[Range[750], Union[FactorInteger[#][[;; , 2]]] == {1, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [1, 3];
```

cp's OEIS Frontend

A036537 Numbers whose number of divisors is a power of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Formula

A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A380922 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(3*s)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375074 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no higher exponents.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375076 Numbers whose prime factorization exponents include at least one 1, at least one 3 and no other exponents.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI