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A225110
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.
6
1, 6, 18, 28, 42, 54, 66, 78, 102, 114, 120, 126, 138, 162, 174, 180, 186, 196, 198, 222, 234, 246, 258, 282, 294, 306, 318, 342, 354, 366, 378, 402, 414, 426, 438, 462, 474, 486, 496, 498, 522, 534, 546, 558, 582, 594, 606, 618, 642, 654, 666, 672, 678, 702, 714
OFFSET
1,2
COMMENTS
By convention, for n = 1, a(1) = 1 with q = 1.
The corresponding q are 1, 4, 4, 6, 4, 4, 4, 4, 4, 4, 16, 4, 4, 4, 4, 15, 4, 6, 4,...
Properties of this sequence:
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.
S = 2 for a majority of n
S = 3 for n = 120, 180, 672, 1890, 8460, 9540,...
S = 4 for n = 30240, 32760, 90720,...
LINKS
EXAMPLE
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.
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.
From Michael De Vlieger, Sep 15 2017: (Start)
Records k and first positions n of records of q that pertain to a(n) for values less than or equal to 10^7:
i k n a(n)
----------------------------
1 1 1 1
2 4 2 6
3 6 4 28
4 10 39 496
5 14 608 8128
6 15 16 180
7 16 11 120
8 17 1543 20482
9 18 2521 33345
10 20 629 8415
11 21 145 1890
12 22 30824 407715
13 24 52 672
14 26 2908 38430
15 28 3034 40128
16 30 1917 25410
17 34 96461 1274100
18 35 1544 20496
19 43 61026 806190
20 45 7839 103530
21 54 5512 72800
22 58 74184 979992
23 69 6871 90720
24 77 270202 3571050
25 80 39625 523776
26 96 2284 30240
27 216 164870 2178540
(End)
MAPLE
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:
MATHEMATICA
Select[Range@ 714, Function[n, AnyTrue[If[n > 1, Rest@ #, #] &@ FoldList[Plus, 1/Divisors@ n], IntegerQ]] (* Michael De Vlieger, Sep 15 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Apr 28 2013
STATUS
approved