A133780 -id:A133780 - cp's OEIS frontend

A133779 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.

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

Offset: 1

Leroy Quet, Sep 23 2007

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).

			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
...

Michael De Vlieger, Table of n, a(n) for n = 1..12248 (Rows 1 <= n <= 2000).
Index entries for sequences related to divisors of numbers

Cf. A133780, A132881, A132882.

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

Table[Select[Divisors@ n, NoneTrue[# + {-1 + 2 Boole[# == 1], 1}, Divisible[n, #] &] &] /. {} -> {0}, {n, 36}] // Flatten (* Michael De Vlieger, Aug 19 2017 *)

More terms from Emeric Deutsch, Oct 02 2007

Extended by Ray Chandler, Jun 24 2008

A133829 a(n) = the largest "non-isolated divisor" of 2n. A positive divisor k of n is non-isolated if k-1 or k+1 also divides n.

Original entry on oeis.org

2, 2, 3, 2, 2, 4, 2, 2, 3, 5, 2, 4, 2, 2, 6, 2, 2, 4, 2, 5, 7, 2, 2, 4, 2, 2, 3, 8, 2, 6, 2, 2, 3, 2, 2, 9, 2, 2, 3, 5, 2, 7, 2, 2, 10, 2, 2, 4, 2, 5, 3, 2, 2, 4, 11, 8, 3, 2, 2, 6, 2, 2, 7, 2, 2, 12, 2, 2, 3, 5, 2, 9, 2, 2, 6, 2, 2, 13, 2, 5, 3, 2, 2, 8, 2, 2, 3, 2, 2, 10, 14, 2, 3, 2, 2, 4, 2, 2, 3

Offset: 1

Leroy Quet, Sep 25 2007

nonn

No odd integer has any non-isolated divisors.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A133780, A133828.

A133829 := proc(n) local divs,k,i ; divs := sort(convert(numtheory[divisors](2*n),list)) ; for i from 1 to nops(divs) do k := op(-i,divs) ; if k-1 in divs or k+1 in divs then RETURN(k) ; fi ; od: RETURN(0) ; end: seq(A133829(n),n=1..100) ; # R. J. Mathar, Oct 19 2007

A133829(n) = { n = 2*n; my(m=0); fordiv(n,d,if(!(n%(1+d)) || ((d>1) && !(n%(d-1))), m = max(m,d))); (m); }; \\ Antti Karttunen, Mar 02 2023

More terms from R. J. Mathar, Oct 19 2007

cp's OEIS Frontend

A133779 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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