A321182 - OEIS

%I #14 Nov 23 2020 02:05:59

%S 15,35,45,95,119,143,209,287,319,323,377,527,559,693,779,899,923,989,

%T 1007,1189,1199,1343,1349,1763,1919,2159,2507,2759,2911,3239,3599,

%U 3827,4031,4607,5183,5207,5249,5459,5543,6439,6811,6887,7067,7279,7739,8159,8639,9179

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

%C 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 Mainly squarefree numbers, which are a subset of A242152, apart from some sporadic terms: 45, 693, 6811, 17296, 24016, 71753, 1165669, etc.

%C A230164 is a subsequence (ratio equal to 1).

%H Amiram Eldar, <a href="/A321182/b321182.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 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 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 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 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 (PARI) ard(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]); \\ A003415

%o isok(n) = (n>1) && !isprime(n) && (frac(sigma(n)/ard(n)) == 0); \\ _Michel Marcus_, Oct 30 2018

%Y Cf. A000203, A003415, A230164, A242152.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Oct 29 2018