A094529 -id:A094529

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A094530

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..39.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A091012, A094529.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 07 2004

STATUS

approved

A270571

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A091011, A094529, A094530.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Mar 19 2016

EXTENSIONS

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

STATUS

approved

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