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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

Cf. A000005, A000203, A007691.

Sequence in context: A028558 A140450 A157800 * A077660 A240991 A304050

Adjacent sequences:  A225107 A225108 A225109 * A225111 A225112 A225113

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 28 2013

STATUS

approved

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Last modified September 22 03:15 EDT 2018. Contains 315267 sequences. (Running on oeis4.)