A091010 -id:A091010 - cp's OEIS frontend

A091009 Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 11, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 10, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

a(A091014(n))=n and a(m)<>n for m<=A091014(n);
a(A091010(n))=0; a(A091011(n))>0; a(A091012(n))=1; a(A091013(n))>1.
Number of pairs (x,y) of divisors of n with xAntti Karttunen, Sep 10 2018

			a(30)=4, as there are exactly 4 triples of divisors with the defining property: (1,2,3), (1,3,5), (2,6,10) and (5,10,15).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. also A094518.

Array[Count[Subsets[#, {3}], ?(#2 - #1 == #3 - #2 & @@ # &)] &@ Divisors@ # &, 105] (* _Michael De Vlieger, Sep 10 2018 *)

A091009(n) = if(1==n,0,my(d=divisors(n),c=0); for(i=1,(#d-1),for(j=(i+1),#d,if(!(n%(d[j]+(d[j]-d[i]))),c++))); (c)); \\ Antti Karttunen, Sep 10 2018

Definition clarified by Antti Karttunen, Sep 10 2018

A091011 Numbers having at least one divisor, d, such that for some x, d-x and d+x are also divisors.

Original entry on oeis.org

6, 12, 15, 18, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 66, 72, 75, 78, 80, 84, 90, 91, 96, 102, 105, 108, 112, 114, 120, 126, 132, 135, 138, 140, 144, 150, 153, 156, 160, 162, 165, 168, 174, 180, 182, 186, 190, 192, 195, 196, 198, 200, 204, 210, 216, 220

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) > 0; complement of A091010.

Numbers k with at least one pair of divisors, (d1,d2), with d1 < d2, whose (integer) average divides k. - Wesley Ivan Hurt, Aug 23 2020

Cf. A091009, A091010, A337331.

Table[If[Sum[Sum[(1 - Ceiling[(i + k)/2] + Floor[(i + k)/2]) (1 - Ceiling[2 n/(i + k)] + Floor[2 n/(i + k)]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}] > 0, n, {}], {n, 200}] // Flatten (* Wesley Ivan Hurt, Aug 23 2020 *)

A091013 Numbers having more than one divisor d such that for some x also d-x and d+x are divisors.

Original entry on oeis.org

12, 18, 24, 30, 36, 42, 45, 48, 54, 56, 60, 66, 72, 75, 78, 80, 84, 90, 96, 102, 105, 108, 112, 114, 120, 126, 132, 135, 138, 140, 144, 150, 156, 160, 162, 165, 168, 174, 180, 182, 186, 192, 195, 196, 198, 200, 204, 210, 216, 222, 224, 225, 228, 231, 234, 240

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) > 1.

Cf. A091012, A091011, A091010.

A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.

Original entry on oeis.org

1, 6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 60, 66, 72, 75, 84, 90, 91, 96, 108, 120, 132, 135, 144, 150, 162, 168, 180, 192, 198, 216, 225, 240, 252, 264, 270, 276, 288, 300, 306, 312, 324, 330, 336, 360, 375, 384, 396, 405, 420, 432, 435, 450, 480, 486, 504

Offset: 1

William Rex Marshall, Oct 24 2013

nonn

Equivalently, the numbers with exactly one divisor that is not in an arithmetic progression of at least three divisors.

Contains p^j*(2*p-1)^k for j,k>=1 if p and 2*p-1 are primes. - Robert Israel, Apr 13 2020

			36 appears in this sequence because its proper divisors are 1, 2, 3, 4, 6, 9, 12 and 18, each of which appears in at least one of the following arithmetic progressions of at least three proper divisors of 36: {1, 2, 3, 4}, {3, 6, 9, 12}, {6, 12, 18}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A091010, A091011.

Contains A033845, A129521.

filter:= proc(n) local S,D,tau,a,b;
  S:= numtheory:-divisors(n) minus {n};
  D:= sort(convert(S,list));
  tau:= nops(D);
  for a from 1 to tau-2 do for b from a+1 to tau-1 do
    if member(2*D[b]-D[a],D) then
      S:= S minus {D[a],D[b],2*D[b]-D[a]};
      if S = {} then return true fi;
    fi
  od od;
  false;
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Apr 13 2020

filterQ[n_] := Module[{S, D, tau, a, b}, S = Most @ Divisors[n]; D = S; tau = Length[D]; For[a = 1, a <= tau - 2, a++, For[b = a + 1, b <= tau - 1, b++, If [MemberQ[D, 2 D[[b]] - D[[a]]], S = S ~Complement~ {D[[a]], D[[b]], 2 D[[b]] - D[[a]]}; If[S == {}, Return[True]]]]]; False];
filterQ[1] = True;
Select[Range[1000], filterQ] (* Jean-François Alcover, Sep 26 2020, after Robert Israel *)

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A091009 Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.

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A091011 Numbers having at least one divisor, d, such that for some x, d-x and d+x are also divisors.

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A091013 Numbers having more than one divisor d such that for some x also d-x and d+x are divisors.

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A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.

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