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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 December 19 03:45 EST 2018. Contains 318245 sequences. (Running on oeis4.)