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A321182 Composite numbers k such that sigma(k)/k' is an integer, where k' is the arithmetic derivative of k.

%I A321182 #14 Nov 23 2020 02:05:59
%S A321182 15,35,45,95,119,143,209,287,319,323,377,527,559,693,779,899,923,989,
%T A321182 1007,1189,1199,1343,1349,1763,1919,2159,2507,2759,2911,3239,3599,
%U A321182 3827,4031,4607,5183,5207,5249,5459,5543,6439,6811,6887,7067,7279,7739,8159,8639,9179
%N A321182 Composite numbers k such that sigma(k)/k' is an integer, where k' is the arithmetic derivative of k.
%C A321182 Alternative definition: Composite numbers such that the ratio between the sum of the reciprocal of their divisors and the sum of the reciprocal of their prime factors, counted with multiplicity, is an integer.
%C A321182 Mainly squarefree numbers, which are a subset of A242152, apart from some sporadic terms: 45, 693, 6811, 17296, 24016, 71753, 1165669, etc.
%C A321182 A230164 is a subsequence (ratio equal to 1).
%H A321182 Amiram Eldar, <a href="/A321182/b321182.txt">Table of n, a(n) for n = 1..10000</a>
%e A321182 Divisors of 45 are 1, 3, 5, 9, 15, 45 and prime factors 3^2, 5: (1/1 + 1/3 + 1/5 + 1/9 + 1/15 + 1 /45)/(1/3 + 1/3 + 1/5) = 2
%e A321182 Divisors of 119 are 1, 7, 17, 119 and prime factors 7, 17: (1/1 + 1/7 + 1/17 + 1 /119)/(1/7 + 1/17) = 6.
%e A321182 Divisors of 552521 are 1, 37, 109, 137, 4033, 5069, 14933, 552521 and prime factors 37, 109, 137: (1/1 + 1/37 + 1/109 + 1/137 + 1 /4033 + 1/5069 + 1/14933 + 1/552521)/(1/37 + 1/109 + 1/137) = 24.
%p A321182 with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 2 to q do if not isprime(n) then a:=add(1/a,a=divisors(n)); b:=ifactors(n)[2]; c:=add(b[k][2]/b[k][1],k=1..nops(b)); if frac(a/c)=0 then print(n); fi; fi; od; end: P(10^7);
%t A321182 Select[Range[4, 10^4], And[CompositeQ@ #, IntegerQ[DivisorSigma[1, #]/If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]]]] &] (* _Michael De Vlieger_, Oct 31 2018 *)
%o A321182 (PARI) ard(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]); \\ A003415
%o A321182 isok(n) = (n>1) && !isprime(n) && (frac(sigma(n)/ard(n)) == 0); \\ _Michel Marcus_, Oct 30 2018
%Y A321182 Cf. A000203, A003415, A230164, A242152.
%K A321182 nonn,easy
%O A321182 1,1
%A A321182 _Paolo P. Lava_, Oct 29 2018