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A255586 Composite k such that Sum_{i=1..t-1} d(i+1)/d(i) is an integer, where d(1), ..., d(t) are the divisors of k in ascending order.

%I A255586 #14 Jan 12 2025 05:00:54
%S A255586 4,8,9,16,18,25,27,32,48,49,50,64,81,98,108,121,125,128,162,169,242,
%T A255586 243,256,289,338,343,361,375,512,529,578,625,722,729,841,961,1024,
%U A255586 1029,1058,1250,1331,1369,1458,1681,1682,1849,1920,1922,2048,2187,2197,2209
%N A255586 Composite k such that Sum_{i=1..t-1} d(i+1)/d(i) is an integer, where d(1), ..., d(t) are the divisors of k in ascending order.
%C A255586 The sequence is infinite because the powers of 2 (A000079) are in the sequence.
%C A255586 The prime powers with even exponents (A056798) are in the sequence.
%C A255586 The cubes of primes (A030078) are in the sequence.
%C A255586 The numbers of the form 2p^2 (A079704) with p prime are in the sequence.
%C A255586 The corresponding integers are 4, 6, 6, 8, 9, 10, 9, 10, 14, 14, 11, 12, 12, 13, 17, 22, 15, 14, 16, 26, 17, 15, 16, 34, 19, ...
%H A255586 Harvey P. Dale, <a href="/A255586/b255586.txt">Table of n, a(n) for n = 1..1000</a>
%e A255586 18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18} and 2/1 + 3/2 + 6/3 + 9/6 + 18/9 = 9 is an integer.
%t A255586 lst={};Do[s=0;Do[s=s+Divisors[n][[i+1]]/Divisors[n][[i]],{i,1,Length[Divisors[n]]-1}];If[IntegerQ[s]&&!PrimeQ[n],AppendTo[lst,n]],{n,2300}];lst
%t A255586 Select[Range[2210],CompositeQ[#]&&IntegerQ[Total[#[[2]]/#[[1]]&/@Partition[ Divisors[ #],2,1]]]&] (* _Harvey P. Dale_, Jul 09 2019 *)
%Y A255586 Cf. A000079, A030078, A079704, A227993, A255585.
%K A255586 nonn
%O A255586 1,1
%A A255586 _Michel Lagneau_, Feb 27 2015