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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.

%I A225110 #20 Jun 13 2025 07:41:22
%S A225110 1,6,18,28,42,54,66,78,102,114,120,126,138,162,174,180,186,196,198,
%T A225110 222,234,246,258,282,294,306,318,342,354,366,378,402,414,426,438,462,
%U A225110 474,486,496,498,522,534,546,558,582,594,606,618,642,654,666,672,678,702,714
%N 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.
%C A225110 By convention, for n = 1, a(1) = 1 with q = 1.
%C A225110 The corresponding q are 1, 4, 4, 6, 4, 4, 4, 4, 4, 4, 16, 4, 4, 4, 4, 15, 4, 6, 4,...
%C A225110 Properties of this sequence:
%C A225110 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 A225110 S = 2 for a majority of n
%C A225110 S = 3 for n = 120, 180, 672, 1890, 8460, 9540,...
%C A225110 S = 4 for n = 30240, 32760, 90720,...
%H A225110 Michael De Vlieger, <a href="/A225110/b225110.txt">Table of n, a(n) for n = 1..10000</a>
%e A225110 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 A225110 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 A225110 From _Michael De Vlieger_, Sep 15 2017: (Start)
%e A225110 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 A225110    i     k        n       a(n)
%e A225110   ----------------------------
%e A225110    1     1        1         1
%e A225110    2     4        2         6
%e A225110    3     6        4        28
%e A225110    4    10       39       496
%e A225110    5    14      608      8128
%e A225110    6    15       16       180
%e A225110    7    16       11       120
%e A225110    8    17     1543     20482
%e A225110    9    18     2521     33345
%e A225110   10    20      629      8415
%e A225110   11    21      145      1890
%e A225110   12    22    30824    407715
%e A225110   13    24       52       672
%e A225110   14    26     2908     38430
%e A225110   15    28     3034     40128
%e A225110   16    30     1917     25410
%e A225110   17    34    96461   1274100
%e A225110   18    35     1544     20496
%e A225110   19    43    61026    806190
%e A225110   20    45     7839    103530
%e A225110   21    54     5512     72800
%e A225110   22    58    74184    979992
%e A225110   23    69     6871     90720
%e A225110   24    77   270202   3571050
%e A225110   25    80    39625    523776
%e A225110   26    96     2284     30240
%e A225110   27   216   164870   2178540
%e A225110 (End)
%p A225110 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 A225110 Select[Range@ 714, Function[n, AnyTrue[If[n > 1, Rest@ #, #] &@ FoldList[Plus, 1/Divisors@ n], IntegerQ]]] (* _Michael De Vlieger_, Sep 15 2017 *)
%o A225110 (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
%Y A225110 Cf. A000005, A000203, A007691.
%K A225110 nonn
%O A225110 1,2
%A A225110 _Michel Lagneau_, Apr 28 2013
%E A225110 Name edited by _Michel Marcus_, Jun 13 2025