A091012 -id:A091012 - 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

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.

A094530 Numbers with exactly one arithmetic progression of four successive divisors (not necessarily consecutive).

Original entry on oeis.org

12, 105, 140, 440, 585, 1729, 3825, 5643, 6380, 7161, 9009, 9867, 10472, 11408, 12025, 13923, 17732, 18705, 19760, 21505, 23715, 25568, 27489, 30272, 36465, 38665, 43472, 52521, 58995, 62307, 62985, 63308, 64467, 65780, 69345, 72105, 81075, 89425, 101065

Offset: 1

Reinhard Zumkeller, May 07 2004

nonn

m*a(n) = A094529(k) for all m>0 and some k.

			Set of divisors of 440: {1,2,4,5,8,10,11,20,22,40,44,55,88,110,220,440}, there is only one arithmetic progression containing at least four terms: {2,5,8,11} = (2+k*3:0<=k<3), therefore 440 is a term.

Cf. A091012, A094529.

oap4Q[n_]:=Length[Select[Subsets[Divisors[n],{4}],Length[Union[ Differences[ #]]]==1&]]==1; Select[Range[102000],oap4Q] (* Harvey P. Dale, Aug 02 2017 *)

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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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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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A094530 Numbers with exactly one arithmetic progression of four successive divisors (not necessarily consecutive).

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Mathematica