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%I #12 Sep 15 2017 21:14:08
%S 1,6,18,28,42,54,66,78,102,114,120,126,138,162,174,180,186,196,198,
%T 222,234,246,258,282,294,306,318,342,354,366,378,402,414,426,438,462,
%U 474,486,496,498,522,534,546,558,582,594,606,618,642,654,666,672,678,702,714
%N Numbers n such that S = Sum_{i = 1..q} 1/d(i) is an integer where d(i) are the divisors of n in increasing order, and q the smallest integer 1 < q <= tau(n) for n > 1.
%C By convention, for n = 1, a(1) = 1 with q = 1.
%C The corresponding q are 1, 4, 4, 6, 4, 4, 4, 4, 4, 4, 16, 4, 4, 4, 4, 15, 4, 6, 4,...
%C Properties of this sequence:
%C q = tau(n) if n = 1, 6, 28, 120, 496,... is a multiply-perfect numbers: n divides sigma(n) (see A007691). This numbers are in the sequence.
%C S = 2 for a majority of n
%C S = 3 for n = 120, 180, 672, 1890, 8460, 9540,...
%C S = 4 for n = 30240, 32760, 90720,...
%H Michael De Vlieger, <a href="/A225110/b225110.txt">Table of n, a(n) for n = 1..10000</a>
%e 18 is in the sequence because the divisors of 18 are 1, 2, 3, 6, 9 and 18 => 1 + 1/2 + 1/3 + 1/6 = 2.
%e 28 is in the sequence because 28 is a multiply-perfect numbers: the divisors are 1, 2, 4, 7, 14, 28 and 1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2.
%e From _Michael De Vlieger_, Sep 15 2017: (Start)
%e Records k and first positions n of records of q that pertain to a(n) for values less than or equal to 10^7:
%e i k n a(n)
%e ----------------------------
%e 1 1 1 1
%e 2 4 2 6
%e 3 6 4 28
%e 4 10 39 496
%e 5 14 608 8128
%e 6 15 16 180
%e 7 16 11 120
%e 8 17 1543 20482
%e 9 18 2521 33345
%e 10 20 629 8415
%e 11 21 145 1890
%e 12 22 30824 407715
%e 13 24 52 672
%e 14 26 2908 38430
%e 15 28 3034 40128
%e 16 30 1917 25410
%e 17 34 96461 1274100
%e 18 35 1544 20496
%e 19 43 61026 806190
%e 20 45 7839 103530
%e 21 54 5512 72800
%e 22 58 74184 979992
%e 23 69 6871 90720
%e 24 77 270202 3571050
%e 25 80 39625 523776
%e 26 96 2284 30240
%e 27 216 164870 2178540
%e (End)
%p with(numtheory): for n from 1 to 1000 do:x:=divisors(n):n1:=nops(x):s:=0:ii:=0:for q from 1 to n1 while(ii=0) do:s:=s+1/x[q]:if s=floor(s) and q>1 then ii:=1: printf(`%d, `,n):else fi:od:od:
%t Select[Range@ 714, Function[n, AnyTrue[If[n > 1, Rest@ #, #] &@ FoldList[Plus, 1/Divisors@ n], IntegerQ]] (* _Michael De Vlieger_, Sep 15 2017 *)
%Y Cf. A000005, A000203, A007691.
%K nonn
%O 1,2
%A _Michel Lagneau_, Apr 28 2013
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