A174905
Numbers with no pair (d,e) of divisors such that d < e < 2*d.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 101, 103, 106
Offset: 1
Cf.
A000040,
A000961,
A001248,
A005279,
A030078,
A030514,
A129511,
A174903,
A237271,
A237593,
A241008,
A241010.
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a174905 n = a174905_list !! (n-1)
a174905_list = filter ((== 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014
-
filter:= proc(n)
local d,q;
d:= numtheory:-divisors(n);
min(seq(d[i+1]/d[i],i=1..nops(d)-1)) >= 2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 08 2014
-
(* it suffices to test adjacent divisors *)
a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]
(* Hartmut F. W. Hoft, Aug 07 2014 *)
Select[Range[106], !MatchQ[Divisors[#], {_, d_, e_, _} /; e < 2d]& ] (* Jean-François Alcover, Jan 31 2018 *)
A129510
Number of distinct differences between pairs of distinct divisors of n.
Original entry on oeis.org
0, 1, 1, 3, 1, 5, 1, 6, 3, 6, 1, 10, 1, 6, 5, 10, 1, 13, 1, 13, 6, 6, 1, 18, 3, 6, 6, 14, 1, 19, 1, 15, 6, 6, 6, 24, 1, 6, 6, 22, 1, 23, 1, 15, 12, 6, 1, 30, 3, 15, 6, 15, 1, 25, 6, 23, 6, 6, 1, 37, 1, 6, 13, 21, 6, 25, 1, 15, 6, 24, 1, 40, 1, 6, 13, 15, 6, 26, 1, 34, 10, 6, 1, 45, 6, 6, 6, 26
Offset: 1
n=44, set of divisors of 44 = {1,2,4,11,22,44}:
44-22=22, 44-11=33, 44-4=40, 44-2=42, 44-1=41,
22-11=11, 22-4=18, 22-2=20, 22-1=21,
11-4=7, 11-2=9, 11-1=10, 4-2=2, 4-1=3, 2-1=1,
a(44) = #{1,2,3,7,9,10,11,18,20,21,22,33,40,41,42} = 15;
n=45, set of divisors of 45 = {1,3,5,9,15,45}:
45-15=30, 45-9=36, 45-5=40, 45-3=42, 45-1=44,
15-9=6, 15-5=10, 15-3=12, 15-1=14,
9-5=4, 9-3=6, 9-1=8, 5-3=2, 5-1=4, 3-1=2,
a(45) = #{2,4,6,8,10,12,14,30,36,40,42,44} = 12.
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a[n_]:=Length[Union[Flatten[Differences/@Subsets[Divisors[n],{2}]]]];Table[a[n],{n,88}] (* James C. McMahon, Jan 21 2025 *)
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a(n)=my(d=divisors(n),v=List()); for(i=1,#d-1,for(j=i+1,#d, listput(v,d[j]-d[i]))); #Set(v) \\ Charles R Greathouse IV, Aug 26 2015
A129512
Numbers with at least two pairs of distinct divisors having equal differences.
Original entry on oeis.org
6, 12, 15, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 138, 140, 144, 150, 153, 156, 160, 162, 165, 168, 174, 176, 180, 182, 186, 189, 190, 192, 195
Offset: 1
See example for a(12) = 45 in A129510.
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import Data.List.Ordered (minus)
a129512 n = a129512_list !! (n-1)
a129512_list = minus [1..] a129511_list
-- Reinhard Zumkeller, Aug 10 2015
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q[k_] := Count[Tally[Differences /@ Subsets[Divisors[k], {2}] // Flatten][[;; , 2]], ?(# > 1 &)] > 0; Select[Range[200], q] (* _Amiram Eldar, Jan 27 2025 *)
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is(n)=my(d=divisors(n)); for(i=1,#d-2, for(j=i+1,#d-1, for(k=1,#d, if(i!=k && setsearch(d, d[j]-d[i]+d[k]), return(1))))); 0 \\ Charles R Greathouse IV, Aug 26 2015
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