A132881 -id:A132881 - 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

A243917 Number of non-twin divisors of n.

Original entry on oeis.org

1, 2, 0, 1, 2, 2, 2, 2, 1, 4, 2, 1, 2, 4, 1, 3, 2, 4, 2, 4, 2, 4, 2, 2, 3, 4, 2, 4, 2, 5, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 3, 4, 2, 4, 3, 6, 2, 4, 2, 6, 4, 6, 2, 4, 2, 4, 2, 4, 2, 5, 4, 6, 2, 4, 2, 6, 2, 6, 2, 4, 3, 4, 4, 6, 2, 6, 3, 4, 2, 5, 4, 4, 2, 6, 2, 9, 4, 4, 2, 4, 4, 6

Offset: 1

Juri-Stepan Gerasimov, Jun 15 2014

nonn

A divisor k of n is non-twin if neither the positive values of k - 2 nor k + 2 divide n.

			The positive divisors of 12 are: 1, 2, 3, 4, 6, 12. Of these, 1 and 3 are twin divisors, 2, 4 and 6 are also twin divisors. The unique non-twin divisor is therefore 12. So a(12) = the number of these divisors, which is 1.

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

Cf. A000005, A132881, A243865.

a243917[n_Integer] := Length[Select[Divisors[n], If[And[# <= 2 || Divisible[n, # - 2] == False, Divisible[n, # + 2] == False], True, False] &]]; a243917 /@ Range[120] (* Michael De Vlieger, Aug 17 2014 *)
nntd[n_]:=Module[{d=Select[Divisors[n],#>2&],t},t=Count[d,?(!Divisible[ n, #-2] && !Divisible[ n,#+2]&)]; If[!Divisible[ n,3],t++]; If[ Divisible[ n,2] && !Divisible[n,4],t++];t]; Array[nntd,100] (* _Harvey P. Dale, May 27 2016 *)

a(n) = sumdiv(n, d, (((d<=2) || (n % (d-2))) && (n % (d+2)))); \\ Michel Marcus, Jun 25 2014

a(n) = A000005(n) - A243865(n).

Corrected by Michel Marcus, Jun 27 2014

A132882 a(n) = the sum of the positive isolated divisors of n.

Original entry on oeis.org

1, 0, 4, 4, 6, 6, 8, 12, 13, 15, 12, 18, 14, 21, 24, 28, 18, 33, 20, 30, 32, 33, 24, 50, 31, 39, 40, 53, 30, 55, 32, 60, 48, 51, 48, 81, 38, 57, 56, 78, 42, 77, 44, 81, 78, 69, 48, 114, 57, 90, 72, 95, 54, 114, 72, 102, 80, 87, 60, 147, 62, 93, 104, 124, 84, 138, 68, 123, 96

Offset: 1

Leroy Quet, Sep 03 2007

nonn

A divisor, d, of n is isolated if neither (d-1) nor (d+1) divides n.

The convention for 1 is that it is an isolated divisor iff n is not even. - Olivier Gérard, Sep 22 2007

			The positive divisors of 56 are: 1,2,4,7,8,14,28,56. Of these, 1 and 2 are adjacent and 7 and 8 are adjacent. The isolated divisors are therefore 4,14, 28,56. So a(56) = 4 +14 +28 +56 = 102.

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

Cf. A132881, A132748.

Table[Plus@@Select[Divisors[n],(#==1||Mod[n,#-1]>0)&&Mod[n,#+1]>0&],{n,1,200}] (* Olivier Gérard, Sep 22 2007 *)

a(n) = A000203(n) - A132748(n), where A000203 is sigma(n), sum of divisors of n.

More terms from Olivier Gérard, Sep 22 2007

A356734 Heinz numbers of integer partitions with at least one neighborless part.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

First differs from A319630 in lacking 1 and having 42 (prime indices: {1,2,4}).
A part x is neighborless if neither x - 1 nor x + 1 are parts.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}

These partitions are counted by A356236.
The singleton case is A356237, counted by A356235 (complement A355393).
The strict case is counted by A356607, complement A356606.
The complement is A356736, counted by A355394.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A286470, A287170 (firsts A066205), A289508, A325160, A328166, A328335, A356231, A356233, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A133950 a(n) = the number of "isolated divisors" of n(n+1)/2. A positive divisor k of n is isolated if neither k-1 nor k+1 divides n.

Original entry on oeis.org

1, 2, 1, 2, 4, 4, 4, 5, 6, 4, 5, 5, 4, 8, 10, 6, 6, 6, 6, 8, 8, 4, 8, 12, 6, 8, 11, 6, 8, 8, 8, 14, 8, 8, 14, 9, 4, 8, 16, 8, 8, 8, 6, 16, 12, 4, 12, 17, 9, 12, 13, 6, 8, 16, 18, 18, 8, 4, 11, 11, 4, 12, 28, 20, 16, 8, 6, 13, 16, 8, 14, 14, 4, 12, 19, 14, 16, 8, 12, 31, 10, 4, 11, 22, 8, 8, 18

Offset: 1

Leroy Quet, Sep 30 2007

nonn

			a(8)=5 because 36 (=8*9/2) has 5 isolated divisors: 6,9,12,18,36.

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A133948, A133949, A063440.

with(numtheory): b:=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: seq(nops(b((1/2)*j*(j+1))),j=1..80); # Emeric Deutsch, Oct 15 2007

a(n) = A063440(n) - A133949(n) = A132881(A000217(n)).

More terms from Emeric Deutsch, Oct 15 2007

Extended by Ray Chandler, Jun 23 2008

A133948 a(n) = the number of "isolated divisors" of n(n+1). A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 6, 5, 4, 6, 6, 4, 8, 12, 6, 7, 7, 6, 13, 9, 4, 10, 16, 8, 11, 16, 8, 9, 9, 8, 16, 11, 12, 21, 12, 4, 11, 22, 10, 9, 9, 8, 24, 15, 4, 14, 21, 14, 17, 16, 8, 11, 22, 22, 23, 11, 4, 16, 16, 4, 17, 32, 22, 23, 11, 8, 18, 22, 12, 16, 16, 4, 17, 26, 20, 21, 11, 14, 37, 15, 4, 16

Offset: 1

Leroy Quet, Sep 30 2007

nonn

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A133947, A133950, A092517.

Table[Length[Divisors[n*(n + 1)]] - Length[Select[Divisors[n*(n + 1)], If[ # > 1, Mod[n*(n + 1), #*(# - 1)] == 0] || Mod[n*(n + 1), #*(# + 1)] == 0 &]], {n, 1, 80}] (* Stefan Steinerberger, Nov 01 2007 *)

a(n) = A092517(n) - A133947(n) = A132881(A002378(n)).

More terms from Stefan Steinerberger, Nov 01 2007

Extended by Ray Chandler, Jun 23 2008

A133997 a(n) = the smallest positive integer with exactly n positive "isolated divisors". A divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

Original entry on oeis.org

1, 3, 9, 15, 36, 45, 126, 96, 144, 120, 324, 240, 336, 432, 360, 480, 672, 864, 720, 840, 1260, 1008, 1080, 1920, 1440, 2040, 1680, 2016, 2160, 3024, 2880, 2520, 4620, 4200, 3360, 5544, 4320, 6048, 6300, 9072, 7200, 11700, 5040, 12096, 7920, 7560, 10800

Offset: 1

Leroy Quet, Oct 01 2007

nonn

Ray Chandler, Table of n, a(n) for n=1..577

Cf. A133996, A132881.

With[{s = KeySort@ PositionIndex@ Table[1 + Count[Rest@ Divisors@ n, ?(NoneTrue[# + {-1, 1}, Divisible[n, #] &] &)], {n, 12000}]}, Function[t, TakeWhile[#, # > 0 &] &@ ReplacePart[t, Map[# -> Lookup[s, #][[1]] &, Keys@ s]]]@ ConstantArray[0, Last@ Keys@ s]] (* _Michael De Vlieger, Oct 11 2017 *)

Extended by Ray Chandler, Jun 24 2008

A134187 a(0)=1. a(n) = the number of terms of the sequence (from among terms a(0) through a(n-1)) which equal any "non-isolated divisors" of (2n). A divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 6, 3, 3, 8, 3, 3, 10, 3, 3, 13, 3, 3, 14, 3, 3, 17, 3, 3, 18, 3, 3, 20, 4, 3, 23, 3, 3, 23, 3, 3, 27, 3, 3, 27, 4, 3, 31, 3, 3, 32, 3, 3, 34, 3, 5, 33, 3, 3, 37, 4, 4, 35, 3, 3, 43, 3, 3, 40, 3, 3, 45, 3, 3, 43, 8, 3, 50, 3, 3, 48, 3, 3, 53, 3, 8, 49, 3, 3, 59, 3, 3, 53, 3, 3, 62, 5

Offset: 0

Leroy Quet, Oct 12 2007

nonn

			The positive divisors of 2*12=24 are 1,2,3,4,6,8,12,24. Of these, 1,2,3,4 are the non-isolated divisors of 24. There are 2 terms among the earlier terms of the sequence that equal 1, 1 term that equals 2, 7 terms which equal 3 and 0 terms which equal 4. So a(12) = 2+1+7+0 = 10.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A132747, A132881, A134188.

up_to = 91;
A134187list(up_to) = { my(v=vector(1+up_to)); v[1] = 1; for(n=1,up_to,v[1+n] = sum(k=0,n-1,my(u=v[1+k]); !((2*n)%u) && ((!((2*n)%(1+u))) || ((u>1)&&(!((2*n)%(u-1))))))); (v); };
v134187 = A134187list(up_to);
A134187(n) = v134187[1+n]; \\ Antti Karttunen, Apr 06 2021

Extended by Ray Chandler, Jun 25 2008

A133952 a(n) = the number of "isolated divisors" of n!. 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, 10, 19, 43, 77, 137, 243, 497, 749, 1520, 2518, 3952, 5294, 10628, 14564, 29199, 40855, 60605, 95786, 191700, 242580, 339732, 531896, 677048, 916946, 1834106, 2332346, 4664982, 5528982, 7863685, 12164443, 16422235, 19594843

Offset: 1

Leroy Quet, Sep 30 2007

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..60 (terms 1..50 from Ray Chandler)

Cf. A133951, A027423.

Cf. A000142, A132881.

A133952 := proc(n) local divs,k,i,a ; divs := sort(convert(numtheory[divisors](n!), list)) ; a := 0 ; 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 a := a+1 ; fi ; od: RETURN(a) ; end: for n from 1 do printf("%d,",A133952(n)) ; od: # R. J. Mathar, Oct 19 2007

a(n) = A027423(n) - A133951(n) = A132881(A000142(n)).

Corrected and extended by R. J. Mathar, Oct 19 2007
a(26)-a(35) from Ray Chandler, May 28 2008
a(36)-a(50) from Ray Chandler, Jun 20 2008

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.

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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A243917 Number of non-twin divisors of n.

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A132882 a(n) = the sum of the positive isolated divisors of n.

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A356734 Heinz numbers of integer partitions with at least one neighborless part.

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A133950 a(n) = the number of "isolated divisors" of n(n+1)/2. A positive divisor k of n is isolated if neither k-1 nor k+1 divides n.

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A133948 a(n) = the number of "isolated divisors" of n(n+1). A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

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A133997 a(n) = the smallest positive integer with exactly n positive "isolated divisors". A divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

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A134187 a(0)=1. a(n) = the number of terms of the sequence (from among terms a(0) through a(n-1)) which equal any "non-isolated divisors" of (2n). A divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

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