A133779 -id:A133779 - cp's OEIS frontend

A134321 Positive integers with the same number of non-isolated divisors as isolated divisors. A divisor k of n is non-isolated if k-1 and/or k+1 also divides n. A divisor k of n is isolated if neither k-1 nor k+1 divides n.

8, 10, 14, 18, 22, 24, 26, 34, 38, 40, 46, 56, 58, 60, 62, 72, 74, 82, 84, 86, 94, 106, 110, 118, 122, 132, 134, 142, 146, 156, 158, 166, 178, 182, 194, 202, 206, 210, 214, 218, 220, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362

Offset: 1

Leroy Quet, Oct 20 2007

nonn

Comments from Hugo van der Sanden, Oct 30 2007 and Oct 31 2007: (Start) Almost all the entries are of the form 2p or 2pq where q = 2p +/- 1 (and so p is in A005383 or A005384). The exceptions are: 8 18 24 40 56 60 72 84 132 156 210 220 380 ... with no others up to 2e6, suggesting that this exception list is finite and complete.
See also my comments on A134320. For the present sequence, we see that elements cannot be perfect squares since those have an odd number of divisors.
Thus they must either be oblong numbers with one isolated divisor below the square root (such as the isolated 5 for 110) or non-oblong numbers with all divisors below the square root being non-isolated.
I expect that proving this sequence consists only of the two general classes and the finite, complete list of exceptions describe above is also possible and would use a similar approach to the first case. (End)

			The divisors of 40 are 1,2,4,5,8,10,20,40. Of these, 1,2,4,5 are non-isolated divisors and 8,10,20,40 are isolated divisors. There are the same number of non-isolated divisors (4 in number) as isolated divisors (4 in number), so 40 is in the sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A133779, A134320, A134322, A243932.

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: nops(ISO) end proc: b:=proc(n) if a(n)=tau(n)-a(n) then n else end if end proc: seq(b(n),n=1..300); # Emeric Deutsch, Oct 24 2007

fQ[n_] := Block[{d = Divisors@ n}, Length@ d == 2Length@ Select[d, MemberQ[d, # + 1] || MemberQ[d, # - 1] &]]; Select[ Range@ 400, fQ] (* Robert G. Wilson v, Jun 22 2014 *)

More terms from Emeric Deutsch and Hugo van der Sanden, Oct 24 2007

A133780 Irregular array: n-th row lists the "non-isolated divisors" 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

1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 5, 6, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 4, 5, 1, 2, 3, 6, 7, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 7, 8, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 2, 3, 4, 8, 9, 1

Offset: 1

Leroy Quet, Sep 23 2007

No odd integer has any non-isolated divisors. The number of terms in the n-th row of the array is A132747(2n).

			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 non-isolated divisors of 20 are 1,2,4,5.

Cf. A133779, A132747, A132748.

Extended by Ray Chandler, Jun 24 2008

A133828 a(n) = the smallest "isolated divisor" of n, or 0 if no such divisor exists. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

1, 0, 1, 4, 1, 6, 1, 4, 1, 5, 1, 6, 1, 7, 1, 4, 1, 6, 1, 10, 1, 11, 1, 6, 1, 13, 1, 4, 1, 10, 1, 4, 1, 17, 1, 6, 1, 19, 1, 8, 1, 14, 1, 4, 1, 23, 1, 6, 1, 5, 1, 4, 1, 6, 1, 4, 1, 29, 1, 10, 1, 31, 1, 4, 1, 6, 1, 4, 1, 5, 1, 6, 1, 37, 1, 4, 1, 6, 1, 8, 1, 41, 1, 12, 1, 43, 1, 4, 1, 15, 1, 4, 1, 47, 1, 6, 1

Offset: 1

Leroy Quet, Sep 25 2007

nonn

a(2n-1) = 1 for all positive integers n. 2 has no isolated divisors. a(2) is 0 only as a placeholder.

			a(18)=6 because the isolated divisors of 18 are 6,9 and 18.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A133779, A133829, A134187, A134188.

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, seq(a(j)[1],j=3..80); # Emeric Deutsch, Oct 16 2007
A133828 := proc(n) local divs,k,i ; divs := sort(convert(numtheory[divisors](n),list)) ; for i from 1 to nops(divs) do k := op(i,divs) ; if not k-1 in divs and not k+1 in divs then RETURN(k) ; fi ; od: RETURN(0) ; end: seq(A133828(n),n=1..100) ; # R. J. Mathar, Oct 19 2007

a[n_] := If[OddQ[n], 1, For[d = 2, d <= n, d++, If[Divisible[n, d] && !Divisible[n, d-1] && !Divisible[n, d+1], Return[d]]]] /. Null -> 0;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2024 *)

A133828(n) = if(n%2,1,fordiv(n,d,if((d>1)&&(n%(d-1))&&(n%(d+1)), return(d))); (0)); \\ Antti Karttunen, Apr 01 2021

More terms from Emeric Deutsch and R. J. Mathar, Oct 16 2007

A134320 Positive integers with more non-isolated divisors than isolated divisors.

Original entry on oeis.org

2, 4, 6, 12, 20, 30, 42, 90

Offset: 1

Leroy Quet, Oct 20 2007

nonn

A divisor k of n is isolated if neither k-1 nor k+1 divides n (see A133779, A132881).
Is this sequence finite? One can show that, with the exception of a(2) = 4, all terms of this sequence must be of the form m*(m+1), oblong numbers, A002378.
Comments from Hugo van der Sanden, Oct 30 2007 and Oct 31 2007: (Start) A quick program to check found no other example up to 3e6, which certainly suggests it is not just finite but complete.
Partial proof: if adjacent integers k, k+1 both divide n then since they are coprime we also have that k(k+1) divides n, so k < sqrt(n).
I.e. the largest non-isolated factor a number can have is ceiling(sqrt(n)).
Since the divisors are symmetrically disposed around the square root, we have: if n is nonsquare, to be in this sequence it must be an oblong number, with all divisors below the square root non-isolated; if n is square, say n = m^2, then we have n divisible by m^2(m-1), so we require m-1 = 1.
So the only square entry is n = 4.
It remains to prove that there is no oblong number greater than 9*10 that avoids isolated divisors below the square root. (End)

			The divisors of 42 are 1,2,3,6,7,14,21,42. Of these, 1,2,3,6,7 are non-isolated divisors and 14,21,42 are isolated divisors. There are more non-isolated divisors (5 in number) than isolated divisors (3 in number), so 42 is in the sequence.

Cf. A134321, A134322.

A134338 a(n) = product of the "isolated divisors" of n. A divisor k of n is isolated if neither k-1 nor k+1 divides n.

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 7, 32, 27, 50, 11, 72, 13, 98, 225, 512, 17, 972, 19, 200, 441, 242, 23, 13824, 125, 338, 729, 10976, 29, 4500, 31, 16384, 1089, 578, 1225, 419904, 37, 722, 1521, 64000, 41, 12348, 43, 42592, 91125, 1058, 47, 10616832, 343, 62500, 2601

Offset: 1

Leroy Quet, Oct 21 2007

nonn

2 has no isolated divisors. So a(2) is 1.

			The divisors of 20 are 1, 2, 4, 5, 10, 20. Of these, 10 and 20 are the isolated divisors. So a(20) = 10*20 = 200.

Cf. A133779, A132881, A134339.

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: product(ISO[j],j=1..nops(ISO)) end proc: seq(a(n),n=1..50); # Emeric Deutsch, Oct 24 2007

isoDivs[n_] := Module[{dn = Divisors[n]}, Complement[dn, Union[Flatten[Select[Partition[dn, 2, 1], #[[2]] - #[[1]] == 1 &]]]]]; Table[Times@@isoDivs[i], {i, 60}] (* Harvey P. Dale, Jan 09 2011 *)

a(2n-1) = A007955(2n-1); a(2n) = A007955(2n) / A134339(n). - Ray Chandler

More terms from Emeric Deutsch, Oct 24 2007

Extended by Ray Chandler, Jun 24 2008

A131847 Positive integers n which each have at least one "isolated isolated divisor". An isolated isolated divisor is an isolated divisor that is not adjacent to any other isolated divisors in the list of all positive divisors of n. A divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

1, 4, 6, 56, 72, 110, 112, 132, 144, 156, 182, 210, 216, 224, 288, 380, 392, 432, 440, 448, 506, 550, 576, 616, 648, 720, 728, 780, 784, 792, 864, 880, 896, 924, 936, 952, 1050, 1064, 1152, 1200, 1210, 1224, 1232, 1260, 1274, 1288, 1296, 1320, 1368, 1404

Offset: 1

Leroy Quet, Oct 04 2007

nonn

			The positive divisors of 56 are: 1,2,4,7,8,14,28,56. Of these, 1 and 2 differ by 1 and 7 and 8 differ by 1. Therefore the isolated divisors of 56 are 4,14,28,56. But 4 is not next to any isolated divisors in the list of all positive divisors of 56. (4 is next to 2 and 7, neither of which is isolated.) So 4 is an isolated isolated divisor of 56 and 56, therefore has at least one isolated isolated divisor.

Cf. A133779.

isIso := proc(k,divs) if not k-1 in divs and not k+1 in divs then true ; else false ; fi ; end: isA131847 := proc(n) local divs,i,isos ; divs := convert(numtheory[divisors](n),list) ; isos := [] ; for i from 1 to nops(divs) do isos := [op(isos),isIso(op(i,divs),divs)] ; od: if nops(isos) = 1 then RETURN(true) ; fi ; if op(1,isos) = true and op(2,isos) = false then RETURN(true) ; fi ; for i from 2 to nops(isos)-1 do if op(i,isos) = true and op(i-1,isos)=false and op(i+1,isos) = false then RETURN(true) ; fi ; od: if op(-1,isos) = true and op(-2,isos) = false then RETURN(true) ; fi ; RETURN(false) ; end: for n from 1 to 1500 do if isA131847(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Oct 24 2007

More terms from R. J. Mathar, Oct 24 2007

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A134321 Positive integers with the same number of non-isolated divisors as isolated divisors. A divisor k of n is non-isolated if k-1 and/or k+1 also divides n. A divisor k of n is isolated if neither k-1 nor k+1 divides n.

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A133780 Irregular array: n-th row lists the "non-isolated divisors" of (2n). A positive divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

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A133828 a(n) = the smallest "isolated divisor" of n, or 0 if no such divisor exists. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

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A134320 Positive integers with more non-isolated divisors than isolated divisors.

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A134338 a(n) = product of the "isolated divisors" of n. A divisor k of n is isolated if neither k-1 nor k+1 divides n.

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