A048108 -id:A048108 - cp's OEIS frontend

A072066 Exceptional (or extraordinary) numbers: m such that A005179(m) < A037019(m).

8, 16, 24, 32, 48, 64, 72, 80, 96, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848

Offset: 1

David Wasserman, Jun 12 2002

nonn

Brown shows that this sequence has density 0 and is a subsequence of A013929. Mei shows that in fact it is a subsequence of A048108. - Charles R Greathouse IV, Jun 07 2013
Not a subsequence of A025487: 80, 108, 112, etc. are not the product of primorials. - Charles R Greathouse IV, Jun 07 2013
The product of any exceptional numbers is an exceptional number. - Thomas Ordowski, Jun 14 2015
Grost proved that p^k is in the sequence if and only if 2^p < prime(k), where p is a prime. - Thomas Ordowski, Jun 15 2015
Only very few of the initial terms, {108, 162, 243, 324, 486, 729, ...} are not multiples of 8. Note that the 2nd to 6th in this list (and certainly more) equal 81*k = (10 + 1/8)*a(n) with n = 2, 3, 4, 5, 7, ... - M. F. Hasler, Jun 15 2022

			m=8 is a term: A005179(8) = 2^3 * 3 = 24 < 30 = 2^1 * 3^1 * 5^1 = A037019(8). - _Jon E. Schoenfield_, Mar 18 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Ron Brown, The minimal number with a given number of divisors, Journal of Number Theory 116:1 (2006), pp. 150-158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
Shu-Yuan Mei, A new class of ordinary integers, video summary of article.
Shu-Yuan Mei, A new class of ordinary integers, Journal of Number Theory, Volume 133, Issue 10, October 2013, Pages 3559-3564.
Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.

Cf. A005179, A037019.

select( {is_A072066(n)=A005179(n)<A037019(n)}, [1..9999]) \\ M. F. Hasler, Oct 14 2014, updated Jun 15 2022

Links updated by Michel Marcus and M. F. Hasler, Oct 14 2014

A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

Original entry on oeis.org

8, 24, 27, 36, 40, 54, 56, 88, 100, 104, 120, 125, 135, 136, 152, 168, 180, 184, 189, 196, 225, 232, 248, 250, 252, 264, 270, 280, 296, 297, 300, 312, 328, 343, 344, 351, 375, 376, 378, 396, 408, 424, 440, 441, 450, 456, 459, 468, 472, 484, 488

Offset: 1

Harvey P. Dale, Sep 08 2011

nonn

From Amiram Eldar, Nov 07 2020: (Start)
Numbers whose powerful part (A057521) is either a cube of a prime (A030078) or a square of a squarefree semiprime (A085986).
The asymptotic density of this sequence is (6/Pi^2) * (Sum_{p prime} 1/(p^2*(p+1)) + Sum_{p=4} (-1)^(k+1)*(k-1)*P(k) + (Sum_{k>=2} (-1)^k*P(k))^2)/2 = 0.0963023158..., where P is the prime zeta function. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
Index entries for sequences computed from exponents in factorization of n

Subsequence of A048108.
Subsequences: A030078 and A085986.
Cf. A001221, A001222, A025487, A057521, A060687, A195069, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A046660, A257851, A261256, A264959.

a195086 n = a195086_list !! (n-1)
a195086_list = filter ((== 2) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[500],PrimeOmega[#]-PrimeNu[#]==2&]

is(n)=bigomega(n)-omega(n)==2 \\ Charles R Greathouse IV, Sep 14 2015

is(n)=my(f=factor(n)[,2]); vecsum(f)==#f+2 \\ Charles R Greathouse IV, Aug 01 2016

A001222(a(n)) - A001221(a(n)) = 2.

A046660(a(n)) = 2. - Reinhard Zumkeller, Nov 29 2015

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).

Original entry on oeis.org

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.

A048106 Number of unitary divisors of n (A034444) - number of non-unitary divisors of n (A048105).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 0, 1, 4, 2, 2, 2, 4, 4, -1, 2, 2, 2, 2, 4, 4, 2, 0, 1, 4, 0, 2, 2, 8, 2, -2, 4, 4, 4, -1, 2, 4, 4, 0, 2, 8, 2, 2, 2, 4, 2, -2, 1, 2, 4, 2, 2, 0, 4, 0, 4, 4, 2, 4, 2, 4, 2, -3, 4, 8, 2, 2, 4, 8, 2, -4, 2, 4, 2, 2, 4, 8, 2, -2, -1, 4, 2, 4, 4, 4, 4, 0, 2, 4, 4, 2, 4, 4, 4, -4, 2, 2, 2

Offset: 1

Labos Elemer

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A008683, A034444, A048105.
Cf. A048107, A048108, A048109, A048111.
Cf. A001620, A306016.

Table[2^(1 + PrimeNu@ n) - DivisorSigma[0, n], {n, 99}] (* Michael De Vlieger, Aug 01 2017 *)

A048106(n) = (2^(1+omega(n)) - numdiv(n)); \\ Antti Karttunen, May 25 2017

from sympy import divisor_count, primefactors
def a(n): return 1 if n==1 else 2**(1 + len(primefactors(n))) - divisor_count(n) # Indranil Ghosh, May 25 2017

a(n) = 2^(1+omega(n)) - d(n) = 2^(1+A001221(n)) - A000005(n).
a(n) = -Sum_{ d divides n } (-1)^mu(d). - Vladeta Jovovic, Jan 24 2002
From Amiram Eldar, Dec 09 2022: (Start)
a(n) > 0 iff n is in A048107.
a(n) < 0 iff n is in A048111.
a(n) <= 0 iff n is in A048108.
a(n) = 0 iff n is in A048109.
Dirichlet g.f: zeta(s)^2*(2/zeta(2*s) - 1).
Sum_{k=1..n} a(k) ~ (12/Pi^2 - 1)*n*log(n) + ((12/Pi^2-1)*(2*gamma-1) - (24/Pi^2)*zeta'(2)/zeta(2))*n, where gamma is Euler's constant (A001620). (End)

A062171 Number of non-unitary divisors of n (A048105) > number of distinct prime divisors of n (A001221).

Original entry on oeis.org

8, 16, 24, 27, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 104, 108, 112, 120, 125, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 228, 232, 234, 240

Offset: 1

Jason Earls, Aug 28 2001

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048105, A001221, A048108.

nud(n) = numdiv(n)-2^omega(n); j=[]; for(n=1,500, if(nud(n)>omega(n),j=concat(j,n))); j

is(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1)>2^#f+#f \\ Charles R Greathouse IV, Sep 18 2015

Offset corrected by Charles R Greathouse IV, Sep 18 2015

cp's OEIS Frontend

A072066 Exceptional (or extraordinary) numbers: m such that A005179(m) < A037019(m).

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A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

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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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A048106 Number of unitary divisors of n (A034444) - number of non-unitary divisors of n (A048105).

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Author

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Programs

Mathematica

PARI

Python

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A062171 Number of non-unitary divisors of n (A048105) > number of distinct prime divisors of n (A001221).

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Author

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Programs

PARI

PARI

Extensions