A059956 -id:A059956 - cp's OEIS frontend

A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86

Offset: 1

Labos Elemer

nonn

Numbers with at most one 2 and no 3s or higher in their prime exponents. - Charles R Greathouse IV, Aug 25 2016

A disjoint union of A005117 and A060687. The asymptotic density of this sequence is (6/Pi^2) * (1 + Sum_{p prime} 1/(p*(p+1))) = A059956 * (1 + A179119) = A059956 + A271971 = 0.8086828238... - Amiram Eldar, Nov 07 2020

			n = 420 = 2*2*3*5*7, 4 distinct prime factors, 24 divisors of which 16 are unitary and 8 are not; ud(n) > nud(n) and 2^(4+1) = 32 is larger than d, the number of divisors.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Complement of A048108.
A072357 is a subsequence.
Cf. A000005, A001221, A005117, A034444, A048105, A059956, A060687, A179119, A271971, A325973.

Select[Range[500], 2^(1 + PrimeNu[#]) > DivisorSigma[0, #] &] (* G. C. Greubel, May 05 2017 *)

is(n)=my(f=factor(n)[, 2], t); for(i=1, #f, if(f[i]>1, if(t||f[i]>2, return(0), t=1))); 1 \\ Charles R Greathouse IV, Sep 17 2015

is(n)=n==1 || factorback(factor(n)[,2])<3 \\ Charles R Greathouse IV, Aug 25 2016

Numbers for which 2^(r(n)+1) > d(n), where r = A001221, d = A000005.

A048111 Number of unitary divisors of n (A034444) < number of non-unitary divisors of n (A048105).

Original entry on oeis.org

16, 32, 36, 48, 64, 72, 80, 81, 96, 100, 108, 112, 128, 144, 160, 162, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 243, 252, 256, 272, 288, 300, 304, 320, 324, 336, 352, 360, 368, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 464, 468, 480, 484

Offset: 1

Labos Elemer

nonn

Numbers n that are expressible as a product of 2 "nonsquarefree" numbers (i.e., there are 2 integers x,y in A001694 such that n = xy). - Benoit Cloitre, Jan 01 2003
Also numbers having more than one square divisor > 1: A046951(a(n)) > 2. - Reinhard Zumkeller, Apr 08 2003
The asymptotic density of this sequence is 1 - (6/Pi^2)*(1 + Sum_{n>=1} 1/prime(n)^2) = 1 - A059956 * (1 + A085548) = 0.1171394347594477824... . - Amiram Eldar, Sep 25 2022

			36 is in the sequence since the number of its unitary divisors, {1, 4, 9, 36} is 4 which is smaller than 5, the number of its non-unitary divisors, {2, 3, 6, 12, 18}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A034444, A048105, A048109, A082293, A013929, A082294, A082295.

Cf. A085548, A059956.

Select[Range[484], DivisorSigma[0, #] > 2^(PrimeNu[#]+1) &] (* Amiram Eldar, Jun 11 2019 *)

is(n)=my(f=factor(n)[,2],t); for(i=1,#f,if(f[i]>1, if(t||f[i]>3, return(1), t=1))); 0 \\ Charles R Greathouse IV, Sep 17 2015

A000005(a(n)) > 2^(1 + A001221(a(n))).

A195069 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 10.

Original entry on oeis.org

2048, 6144, 9216, 10240, 13824, 14336, 20736, 22528, 25600, 26624, 30720, 31104, 34816, 38912, 43008, 46080, 46656, 47104, 50176, 59392, 63488, 64000, 64512, 67584, 69120, 69984, 71680, 75776, 76800, 79872, 83968, 88064, 96256, 96768, 101376, 103680, 104448

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0003698..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 25 2024

			14336 = 2^11 * 7^1, so it has 12 prime factors (counted with multiplicity) and 2 distinct prime factors, and 12-2 = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A025487, A060687, A195087, A195088, A195089, A195090, A195091, A195092, A195093.

Cf. A046660, A059956, A112526, A257851, A261256, A264959.

a195069 n = a195069_list !! (n-1)
a195069_list = filter ((== 10) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

op(select(n->bigomega(n)-nops(factorset(n))=10, [$1..104448])); # Paolo P. Lava, Jul 03 2018

Select[Range[200000], PrimeOmega[#] - PrimeNu[#] == 10&]

isok(n) = bigomega(n) - omega(n) == 10; \\ Michel Marcus, Jul 03 2018

A046660(a(n)) = 10. - Reinhard Zumkeller, Nov 29 2015

A195088 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 4.

Original entry on oeis.org

32, 96, 144, 160, 216, 224, 243, 324, 352, 400, 416, 480, 486, 544, 608, 672, 720, 736, 784, 928, 992, 1000, 1008, 1056, 1080, 1120, 1184, 1200, 1215, 1248, 1312, 1376, 1504, 1512, 1584, 1620, 1632, 1696, 1701, 1760, 1800, 1824, 1872, 1888, 1936, 1952

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0237194..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195089, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195088 n = a195088_list !! (n-1)
a195088_list = filter ((== 4) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[2000],PrimeOmega[#]-PrimeNu[#]==4&]

is(n)=bigomega(n)-omega(n)==4 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 4. - Reinhard Zumkeller, Nov 29 2015

A195090 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 6.

Original entry on oeis.org

128, 384, 576, 640, 864, 896, 1296, 1408, 1600, 1664, 1920, 1944, 2176, 2187, 2432, 2688, 2880, 2916, 2944, 3136, 3712, 3968, 4000, 4032, 4224, 4320, 4374, 4480, 4736, 4800, 4992, 5248, 5504, 6016, 6048, 6336, 6480, 6528, 6784

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0059189..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195090 n = a195090_list !! (n-1)
a195090_list = filter ((== 6) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

op(select(n->bigomega(n)-nops(factorset(n))=6, [$1..6784])); # Paolo P. Lava, Jul 03 2018

Select[Range[7000],PrimeOmega[#]-PrimeNu[#]==6&]

is(n)=bigomega(n)-omega(n)==6 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 6. - Reinhard Zumkeller, Nov 29 2015

A195092 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 8.

Original entry on oeis.org

512, 1536, 2304, 2560, 3456, 3584, 5184, 5632, 6400, 6656, 7680, 7776, 8704, 9728, 10752, 11520, 11664, 11776, 12544, 14848, 15872, 16000, 16128, 16896, 17280, 17496, 17920, 18944, 19200, 19683, 19968, 20992, 22016, 24064, 24192

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0014793..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195091, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195092 n = a195092_list !! (n-1)
a195092_list = filter ((== 8) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[25000],PrimeOmega[#]-PrimeNu[#]==8&]

is(n)=bigomega(n)-omega(n)==8 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 8. - Reinhard Zumkeller, Nov 29 2015

A195093 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 9.

Original entry on oeis.org

1024, 3072, 4608, 5120, 6912, 7168, 10368, 11264, 12800, 13312, 15360, 15552, 17408, 19456, 21504, 23040, 23328, 23552, 25088, 29696, 31744, 32000, 32256, 33792, 34560, 34992, 35840, 37888, 38400, 39936, 41984, 44032, 48128, 48384

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0007396..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195091, A195092.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195093 n = a195093_list !! (n-1)
a195093_list = filter ((== 9) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[50000],PrimeOmega[#]-PrimeNu[#]==9&]

is(n)=bigomega(n)-omega(n)==9 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 9. - Reinhard Zumkeller, Nov 29 2015

A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum multiplicity is one greater than their minimum (counted by A325279).

The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... . - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  28: {1,1,4}
  44: {1,1,5}
  45: {2,2,3}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}
  99: {2,2,5}

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Positions of 1's in A062977. Supersequence of A054753, A096156.
Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.
Cf. A059956, A088453.

Select[Range[100],Max@@FactorInteger[#][[All,2]]-Min@@FactorInteger[#][[All,2]]==1&]
Select[Range[300],  Min[e = FactorInteger[#][[;; , 2]]] +1 == Max[e] &] (* Amiram Eldar, Jan 30 2023 *)

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023

from sympy import factorint
def ok(n):
    e = sorted(factorint(n).values())
    return n > 1 and max(e) == 1 + min(e)
print([k for k in range(280) if ok(k)]) # Michael S. Branicky, Dec 18 2021

A051903(a(n)) - A051904(a(n)) = 1.

A053462 Number of positive squarefree integers less than 10^n.

Original entry on oeis.org

0, 6, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, 6079270942, 60792710280, 607927102274, 6079271018294, 60792710185947, 607927101854103, 6079271018540405, 60792710185403794, 607927101854022750, 6079271018540280875, 60792710185402613302

Offset: 0

Harvey P. Dale, Aug 01 2001

nonn

			There are 608 squarefree integers smaller than 1000.

Amiram Eldar, Table of n, a(n) for n = 0..36 (from the b-file at A071172; terms 0..20 from Charles R Greathouse IV)
Gérard P. Michon, On the number of squarefree integers not exceeding N.

Cf. A005117, A013928.
Cf. A059956, A063035.
Apart from first two terms, same as A071172.
Binary counterpart is A143658. - Gerard P. Michon, Apr 30 2009

a[n_] := Module[{t=10^n-1}, Sum[MoebiusMu[k]Floor[t/k^2], {k, 1, Sqrt[t]}]]

a(n)=sum(d=1,sqrtint(n=10^n-1), n\d^2*moebius(d)) \\ Charles R Greathouse IV, Nov 14 2012

a(n)=my(s); forsquarefree(d=1,sqrtint(n=10^n-1), s += n\d[1]^2 * moebius(d)); s \\ Charles R Greathouse IV, Jan 08 2018

from math import isqrt
from sympy import mobius
def A053462(n):
    m = 10**n-1
    return sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1)) # Chai Wah Wu, Jun 01 2024

a(n)/10^n = (6/Pi^2)*(1+o(1)), cf. A059956.

a(n) = A071172(n) - [n <= 1] where [] is the Iverson bracket. - Chai Wah Wu, Jun 01 2024

More terms from Dean Hickerson and Vladeta Jovovic, Aug 06 2001
One more term from Jud McCranie, Sep 01 2005
a(0)=0 and a(14)-a(17) from Gerard P. Michon, Apr 30 2009
a(18)-a(20) from Charles R Greathouse IV, Jan 08 2018

A342586 a(n) is the number of pairs (x,y) with 1 <= x, y <= 10^n and gcd(x,y)=1.

Original entry on oeis.org

1, 63, 6087, 608383, 60794971, 6079301507, 607927104783, 60792712854483, 6079271032731815, 607927102346016827, 60792710185772432731, 6079271018566772422279, 607927101854119608051819, 60792710185405797839054887, 6079271018540289787820715707, 607927101854027018957417670303

Offset: 0

Karl-Heinz Hofmann, Mar 16 2021

nonn

Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54. (See link below.)

Joachim von zur Gathen and Jürgen Gerhard, Extract from "3.4. (Non-)Uniqueness of the gcd" chapter, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.
Hugo Pfoertner, Illustration of a(2)=6087.

a(n) = 2*A064018(n) - 1. - Hugo Pfoertner, Mar 16 2021
a(n) = A018805(10^n). - Michel Marcus, Mar 16 2021
Cf. A000010, A059956 (6/Pi^2), A342632, A342841, A343193, A343282.
Related counts of k-tuples:
pairs: A018805, A342632, A342586;
triples: A071778, A342935, A342841;
quadruples: A082540, A343527, A343193;
5-tuples: A343282;
6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

a342586(n)=my(s, m=10^n); forfactored(k=1,m,s+=eulerphi(k)); s*2-1 \\ Bruce Garner, Mar 29 2021

a342586(n)=my(s, m=10^n); forsquarefree(k=1,m,s+=moebius(k)*(m\k[1])^2); s \\ Bruce Garner, Mar 29 2021

import math
for n in range (0,10):
     counter = 0
     for x in range (1, pow(10,n)+1):
        for y in range(1, pow(10,n)+1):
            if math.gcd(y,x) ==  1:
                counter += 1
     print(n, counter)

from functools import lru_cache
@lru_cache(maxsize=None)
def A018805(n):
  if n == 1: return 1
  return n*n - sum(A018805(n//j) for j in range(2, n//2+1)) - (n+1)//2
print([A018805(10**n) for n in range(8)]) # Michael S. Branicky, Mar 18 2021

Lim_{n->infinity} a(n)/10^(2*n) = 6/Pi^2 = 1/zeta(2).

a(10) from Michael S. Branicky, Mar 18 2021
More terms using A064018 from Hugo Pfoertner, Mar 18 2021
Edited by N. J. A. Sloane, Jun 13 2021

cp's OEIS Frontend

A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

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A048111 Number of unitary divisors of n (A034444) < number of non-unitary divisors of n (A048105).

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A195069 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 10.

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A195088 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 4.

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A195090 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 6.

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A195092 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 8.

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A195093 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 9.

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