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

1,2

Comments

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

Numbers k such that 2*A325973(k) = A034448(k)+A048250(k) > A000203(k), or equally, numbers k for which A325973(k) > A325974(k). Proof: for squarefree numbers the left hand side = 2*sigma(k). For k in A060687, i.e., when k = p^2 * r, with r squarefree, A034448(k)+A048250(k) = (p^2 + 1)*sigma(r) + (p + 1)*sigma(r) = (p^2 + p + 2)*sigma(r) = sigma(k) + sigma(r) > sigma(k). But for p^e * r, e >= 3, A034448(k)+A048250(k) = (p^e + 1)*sigma(r) + (p + 1)*sigma(r) = (p^e + p + 2)*sigma(r) < sigma(p^e)*sigma(r) = sigma(k). Likewise if the powerful part of k contains at least two distinct prime factors, then also A034448(k)+A048250(k) < A000203(k). - Antti Karttunen, Oct 05 2025

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Complement of A048108.

A072357 is a subsequence.

Cf. A000005, A001221, A005117, A034444, A034448, A048105, A048250, A059956, A060687, A179119, A271971, A325973, A325974.

Sequence in context: A382063 A382065 A004709 * A342521 A078129 A325368

Adjacent sequences: A048104 A048105 A048106 * A048108 A048109 A048110

Keyword

nonn

Author

Labos Elemer

Status

approved