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A003624 Duffinian numbers: composite numbers k relatively prime to sigma(k).

%I A003624 M3324 #61 Jul 06 2023 10:55:54
%S A003624 4,8,9,16,21,25,27,32,35,36,39,49,50,55,57,63,64,65,75,77,81,85,93,98,
%T A003624 100,111,115,119,121,125,128,129,133,143,144,155,161,169,171,175,183,
%U A003624 185,187,189,201,203,205,209,215,217,219,221,225,235,237,242,243,245,247
%N A003624 Duffinian numbers: composite numbers k relatively prime to sigma(k).
%C A003624 All prime powers greater than 1 are in the sequence. No factorial number can be a term. - _Arkadiusz Wesolowski_, Feb 16 2014
%C A003624 Even terms are in A088827. Any term also in A005153 is either an even square or twice an even square not divisible by 3. - _Jaycob Coleman_, Jun 08 2014
%C A003624 All primes satisfy the second condition since gcd(p, p+1) = 1, thus making this sequence a proper subset of A014567. - _Robert G. Wilson v_, Oct 02 2014
%D A003624 T. Koshy, Elementary number theory with applications, Academic Press, 2002, p. 141, exerc. 6,7,8 and 9.
%D A003624 L. Richard Duffy, The Duffinian numbers, Journal of Recreational Mathematics 12 (1979), pp. 112-115.
%D A003624 Peter Heichelheim, There exist five Duffinian consecutive integers but not six, Journal of Recreational Mathematics 14 (1981-1982), pp. 25-28.
%D A003624 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 64.
%D A003624 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003624 Amiram Eldar, <a href="/A003624/b003624.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A003624 Florian Luca, <a href="http://projecteuclid.org/euclid.mjms/1316032973">On the densities of some subsets of integers</a>, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.
%H A003624 Rose Mary Zbiek, <a href="http://www.kappamuepsilon.org/pages/a/Pentagon/Vol_42_Num_2_Spring_1983.pdf">What can we say about the Duffinian numbers?</a>, The Pentagon 42:2 (1983), pp. 99-109.
%F A003624 A009194(a(n)) * (1 - A010051(a(n))) = 1. - _Reinhard Zumkeller_, Mar 23 2013
%F A003624 a(n) >> n log log log n, see Luca. (Clearly excluding the primes only makes the n-th term larger.) - _Charles R Greathouse IV_, Feb 17 2014
%e A003624 4 is in the sequence since it is not a prime, its divisors 1, 2, and 4 sum to 7, and gcd(7, 4) = 1.
%e A003624 21 is in the sequences since it is not a prime, and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21.
%t A003624 fQ[n_] := n != 1 && !PrimeQ[n] && GCD[n, DivisorSigma[1, n]] == 1; Select[ Range@ 280, fQ]
%o A003624 (PARI) is(n)=gcd(n,sigma(n))==1&&!isprime(n) \\ _Charles R Greathouse IV_, Feb 13 2013
%o A003624 (Haskell)
%o A003624 a003624 n = a003624_list !! (n-1)
%o A003624 a003624_list = filter ((== 1) . a009194) a002808_list
%o A003624 -- _Reinhard Zumkeller_, Mar 23 2013
%o A003624 (Python)
%o A003624 from math import gcd
%o A003624 from itertools import count, islice
%o A003624 from sympy import isprime, divisor_sigma
%o A003624 def A003624_gen(startvalue=2): # generator of terms
%o A003624     return filter(lambda k:not isprime(k) and gcd(k,divisor_sigma(k))==1,count(max(startvalue,2)))
%o A003624 A003624_list = list(islice(A003624_gen(),30)) # _Chai Wah Wu_, Jul 06 2023
%Y A003624 Cf. A000203, A002808, A014567, A025475.
%K A003624 nonn
%O A003624 1,1
%A A003624 _N. J. A. Sloane_, _Mira Bernstein_, Sep 19 1994