A225110 Numbers m such that S = Sum_{i = 1..q} 1/d(i) is an integer where d(i) are the divisors of m in increasing order, and q the smallest integer 1 < q <= tau(m) for m > 1; and a(1) = 1.

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

Author

Michel Lagneau, Apr 28 2013

Keywords

• nonn

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,...

Examples

			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)
		

Links

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

Crossrefs

Cf. A000005, A000203, A007691.

Programs

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 *)

PARI

isok(k) = if (k==1, return(1)); my(d=divisors(k), s=1); for (i=2, #d, s += 1/d[i]; if (denominator(s)==1, return(1));); \\ Michel Marcus, Feb 22 2025

Extensions

Name edited by Michel Marcus, Jun 13 2025