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A133779
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Irregular array: n-th row lists the "isolated divisors" of n. A positive divisor k of n is isolated if neither k-1 nor k+1 divides n.
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7
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1, 0, 1, 3, 4, 1, 5, 6, 1, 7, 4, 8, 1, 3, 9, 5, 10, 1, 11, 6, 12, 1, 13, 7, 14, 1, 3, 5, 15, 4, 8, 16, 1, 17, 6, 9, 18, 1, 19, 10, 20, 1, 3, 7, 21, 11, 22, 1, 23, 6, 8, 12, 24, 1, 5, 25, 13, 26, 1, 3, 9, 27, 4, 7, 14, 28, 1, 29, 10, 15, 30, 1, 31, 4, 8, 16, 32, 1, 3, 11, 33, 17, 34, 1, 5, 7, 35, 6
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OFFSET
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1,4
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COMMENTS
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The second term of the sequence, which corresponds to the second row of the array, is 0 simply as a placeholder, since 2 has no isolated divisors.
The number of terms in the n-th row of the array is A132881(n) (with the exception of row 2, which has 0 elements, but is represented here as 0).
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LINKS
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EXAMPLE
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The positive divisors of 20 are 1,2,4,5,10,20. Of these, 1 and 2 are adjacent and 4 and 5 are adjacent. So the isolated divisors of 20 are 10 and 20.
Triangle begins:
1
-
1,3
4
1,5
6
1,7
4,8
1,3,9
5,10
1,11
6,12
1,13
7,14
1,3,5,15
4,8,16
...
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MAPLE
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with(numtheory): a:=proc(n) local div, ISO, i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1, div)=false and member(div[i]+1, div)=false then ISO:=`union`(ISO, {div[i]}) end if end do end proc: 1; 0; for j from 3 to 30 do seq(a(j)[i], i=1..nops(a(j)))end do; # yields sequence in the form of an array - Emeric Deutsch, Oct 02 2007
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MATHEMATICA
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Table[Select[Divisors@ n, NoneTrue[# + {-1 + 2 Boole[# == 1], 1}, Divisible[n, #] &] &] /. {} -> {0}, {n, 36}] // Flatten (* Michael De Vlieger, Aug 19 2017 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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