Search: a091012 -id:a091012
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A091009
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Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.
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+10
10
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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
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OFFSET
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1,12
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COMMENTS
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Number of pairs (x,y) of divisors of n with x<y such that also 2y-x is a divisor of n. - Antti Karttunen, Sep 10 2018
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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Array[Count[Subsets[#, {3}], _?(#2 - #1 == #3 - #2 & @@ # &)] &@ Divisors@ # &, 105] (* Michael De Vlieger, Sep 10 2018 *)
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PROG
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(PARI) 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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A091013
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Numbers having more than one divisor d such that for some x also d-x and d+x are divisors.
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+10
2
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A094530
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Numbers with exactly one arithmetic progression of four successive divisors (not necessarily consecutive).
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+10
2
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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
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OFFSET
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1,1
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COMMENTS
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m*a(n) = A094529(k) for all m>0 and some k.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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