A091011 -id:A091011 - 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

A091010 Numbers having no divisor d such that also d-x and d+x are divisors for some x.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) = 0; complement of A091011.

A091012 Numbers having exactly one divisor d such that for some x also d-x and d+x are divisors.

Original entry on oeis.org

6, 15, 28, 40, 91, 153, 190, 220, 325, 496, 544, 561, 572, 627, 703, 861, 897, 935, 946, 1012, 1225, 1287, 1292, 1431, 1581, 1610, 1653, 1768, 1891, 2278, 2300, 2465, 2618, 2701, 2967, 3321, 3344, 3496, 3596, 3655, 3952, 4123, 4301, 4324, 4371

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) = 1.

Cf. A091013, A091011.

A094529 Numbers with at least one arithmetic progression of four divisors (not necessarily consecutive).

Original entry on oeis.org

12, 24, 36, 48, 60, 72, 84, 96, 105, 108, 120, 132, 140, 144, 156, 168, 180, 192, 204, 210, 216, 228, 240, 252, 264, 276, 280, 288, 300, 312, 315, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 440, 444, 456, 468, 480, 492, 504, 516, 525, 528, 540, 552, 560

Offset: 1

Reinhard Zumkeller, May 07 2004

nonn

All multiples of terms are also terms.

a(n) = m*A094530(k) for appropriate m and k.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A091011, A094530.

See A270571 for another version.

sd4Q[n_]:=Count[Subsets[Divisors[n],{4}],?(Length[Union[ Differences[ #]]] == 1&)]>0; Select[Range[600],sd4Q] (* _Harvey P. Dale, Mar 19 2016 *)

Edited by Harvey P. Dale and Alois P. Heinz, Mar 18 2016

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.

A270571 Numbers with at least one arithmetic progression of four consecutive divisors.

Original entry on oeis.org

12, 24, 36, 48, 60, 72, 84, 96, 105, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 315, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 525, 528, 540, 552, 564, 576, 588, 600

Offset: 1

Reinhard Zumkeller, Mar 19 2016

nonn

Contrast A094529 where the divisors in arithmetic progression do not have to be consecutive.

			348 is included because its divisors are 1, 2, 3, 4, 6, 12, 29, 58, 87, 116, 174, and 348, and the first four are in arithmetic progression.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A091011, A094529, A094530.

ap4dQ[n_]:=Count[Partition[Divisors[n],4,1],_?(Length[ Union[ Differences[ #]]] == 1&)]>0; Select[ Range[700],ap4dQ]

Edited by Harvey P. Dale and Alois P. Heinz, Mar 19 2016

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 *)

cp's OEIS Frontend

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

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A091010 Numbers having no divisor d such that also d-x and d+x are divisors for some x.

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A091012 Numbers having exactly one divisor d such that for some x also d-x and d+x are divisors.

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A094529 Numbers with at least one arithmetic progression of four divisors (not necessarily consecutive).

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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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A270571 Numbers with at least one arithmetic progression of four consecutive 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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