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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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%I #31 Aug 10 2024 01:29:38

%S 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,

%T 30,31,33,34,35,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,58,

%U 59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84,85,86

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

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

%C 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

%H G. C. Greubel, <a href="/A048107/b048107.txt">Table of n, a(n) for n = 1..5000</a>

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

%e 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.

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

%o (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

%o (PARI) is(n)=n==1 || factorback(factor(n)[,2])<3 \\ _Charles R Greathouse IV_, Aug 25 2016

%Y Complement of A048108.

%Y A072357 is a subsequence.

%Y Cf. A000005, A001221, A005117, A034444, A048105, A059956, A060687, A179119, A271971, A325973.

%K nonn

%O 1,2

%A _Labos Elemer_