A132747 -id:A132747 - cp's OEIS frontend

A133947 a(n) = the number of "non-isolated divisors" of n(n+1). A positive divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

2, 3, 4, 4, 5, 5, 4, 6, 7, 4, 6, 6, 4, 8, 8, 4, 5, 5, 6, 11, 7, 4, 6, 8, 4, 5, 8, 4, 7, 7, 4, 8, 5, 4, 15, 6, 4, 5, 10, 6, 7, 7, 4, 12, 9, 4, 6, 9, 4, 7, 8, 4, 5, 10, 10, 9, 5, 4, 8, 8, 4, 7, 10, 6, 9, 5, 4, 6, 10, 4, 8, 8, 4, 7, 10, 4, 11, 5, 6, 13, 5, 4, 8, 15, 4, 5, 8, 4, 9, 13, 6, 6, 5, 4, 12, 6, 4, 9

Offset: 1

Leroy Quet, Sep 30 2007

nonn

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

Cf. A133948, A133949, A092517.

Table[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) - A133948(n) = A132747(A002378(n)).

More terms from Stefan Steinerberger, Nov 01 2007

Extended by Ray Chandler, Jun 23 2008

A133949 a(n) = the number of "non-isolated divisors" of n(n+1)/2. A positive divisor k of n is non-isolated if either k-1 or k+1 also divides n.

Original entry on oeis.org

0, 0, 3, 2, 0, 0, 2, 4, 0, 0, 3, 3, 0, 0, 6, 2, 0, 0, 2, 8, 0, 0, 4, 6, 0, 0, 5, 2, 0, 0, 2, 6, 0, 0, 10, 3, 0, 0, 8, 4, 0, 0, 2, 8, 0, 0, 4, 7, 0, 0, 3, 2, 0, 0, 6, 6, 0, 0, 5, 5, 0, 0, 8, 4, 0, 0, 2, 3, 0, 0, 4, 4, 0, 0, 5, 2, 0, 0, 4, 9, 0, 0, 5, 10, 0, 0, 6, 2, 0, 0, 4, 3, 0, 0, 10, 4, 0, 0, 8, 2, 0, 0, 2, 13

Offset: 1

Leroy Quet, Sep 30 2007

nonn

a(k) = 0 for k mod 4 == {1,2}. - Ray Chandler

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

Cf. A133947, A133950, A063440.

Table[Length[Select[Divisors[n*(n + 1)/2], If[ # > 1, Mod[n*(n + 1)/2, #*(# - 1)] == 0] || Mod[n*(n + 1)/2, #*(# + 1)] == 0 &]], {n, 1, 80}] (* Stefan Steinerberger, Nov 01 2007 *)

a(n) = A063440(n) - A133950(n) = A132747(A000217(n)).

More terms from Stefan Steinerberger, Nov 01 2007

Extended by Ray Chandler, Jun 23 2008

A133996 a(n) = the smallest positive integer with exactly n positive "non-isolated divisors". A divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

Original entry on oeis.org

2, 6, 12, 30, 60, 90, 180, 360, 630, 420, 840, 1680, 4200, 1260, 2520, 3780, 7560, 15120, 13860, 36960, 49140, 27720, 32760, 65520, 83160, 98280, 196560, 314160, 277200, 491400, 526680, 471240, 360360, 1164240, 942480, 720720, 1081080

Offset: 2

Leroy Quet, Oct 01 2007

nonn

Ray Chandler, Table of n, a(n) for n=2..85

Cf. A133997, A132747.

With[{s = KeySort@ PositionIndex@ Table[1 + Count[Rest@ Divisors@ n, ?(AnyTrue[# + {-1, 1}, Divisible[n, #] &] &)], {n, 1200000}]}, Function[t, TakeWhile[#, # > 0 &] &@ Rest@ 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

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

A328510 Smallest number whose divisors have n non-singleton runs.

Original entry on oeis.org

1, 2, 20, 90, 630, 1260, 3780, 21420, 41580, 128520, 270270, 554400, 706860, 1413720, 2042040, 4324320, 4084080, 9189180, 6126120, 43825320, 12252240, 18378360, 82162080, 36756720, 85765680, 73513440, 183783600, 306306000, 257297040, 563603040, 514594080

Offset: 0

Gus Wiseman, Oct 18 2019

nonn

			The sequence of terms together with their non-singleton runs of divisors begins:
    1: {}
    2: {{1,2}}
   20: {{1,2},{4,5}}
   90: {{1,2,3},{5,6},{9,10}}
  630: {{1,2,3},{5,6,7},{9,10},{14,15}}

Giovanni Resta, Table of n, a(n) for n = 0..54

Equal {1} followed by the positions of first appearances in A328511 (times 2).
The longest run of divisors of n has length A055874.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The number of singleton runs of divisors is A132881.
Cf. A000005, A027750, A033676, A060775, A119313, A132747, A181063, A199970, A328166, A328457.

dv=Table[Length[DeleteCases[Length/@Split[Divisors[n],#2==#1+1&],1]],{n,1000}];
Table[Position[dv,i][[1,1]],{i,Union[dv]}]

Offset changed to 0 and a(10)-a(30) added by Giovanni Resta, Oct 25 2019

A328511 Number of non-singleton runs of divisors of 2n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 18 2019

nonn

			The divisors of 90 have runs: {{1, 2, 3}, {5, 6}, {9, 10}, {15}, {18}, {30}, {45}, {90}}, so a(45) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Run (cards)

Positions of first appearances are A328510.
The longest run of divisors of n has length A055874.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The number of singleton runs of divisors is A132881.
Cf. A000005, A033676, A060775, A119313, A132747, A181063, A199970, A328166, A328457.

f:= proc(n) local D,B,R;
  D:= sort(convert(numtheory:-divisors(2*n),list));
  B:= D[2..-1]-D[1..-2];
  R:= select(j -> (j=1 or B[j-1]>1) and B[j]=1, [$1..nops(B)]);
  nops(R);
end proc:
map(f, [$1..100]); # Robert Israel, Oct 25 2019

Table[Length[DeleteCases[Length/@Split[Divisors[2*n],#2==#1+1&],1]],{n,100}]

A356733 Number of neighborless parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 2, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 1, 2, 2, 0, 0, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 0, 1, 2, 2, 2, 1, 0, 2, 2, 2, 2, 1, 0, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 0, 1, 2, 0, 2, 0, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 0, 2, 2, 2, 2, 2, 0, 1, 2, 2, 2, 1, 1, 1, 2, 0

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

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 prime indices of 42 are {1,2,4}, of which only 4 is neighborless, so a(42) = 1.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 0.
The prime indices of 1300 are {1,1,3,3,6}, with neighborless parts {1,3,6}, so a(1300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are 1 followed by A066205.
Dominated by A287170 (firsts also A066205).
Positions of terms > 0 are A356734.
The complement is counted by A356735.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together prime indices.
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.
A355393 counts partitions w/o a neighborless singleton, complement A356235.
A355394 counts partitions w/o a neighborless part, complement A356236.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Union[primeMS[n]],!MemberQ[primeMS[n],#-1]&&!MemberQ[primeMS[n],#+1]&]],{n,100}]

A356733(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ Antti Karttunen, Jan 28 2025

a(n) = A001221(n) - A356735(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376

Offset: 1

Gus Wiseman, Sep 24 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms and corresponding intervals begin:
        0: ()
        1: (1)
        2: (2)
        4: (3)
        6: (1,2)
        8: (4)
       16: (5)
       20: (2,3)
       32: (6)
       52: (1,2,3)
       64: (7)
       72: (3,4)
      128: (8)
      256: (9)
      272: (4,5)
      328: (2,3,4)
      512: (10)
      840: (1,2,3,4)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
These compositions are counted by A001227.
An unordered version is A073485, non-strict A073491 (complement A073492).
The initial version is A164894, non-strict A356843 (unordered A356845).
The non-strict version is A356841, initial A333217, counted by A107428.
A066311 lists gapless numbers.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0,1000],chQ[stc[#]]&]

cp's OEIS Frontend

A133947 a(n) = the number of "non-isolated divisors" of n(n+1). A positive divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

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A133949 a(n) = the number of "non-isolated divisors" of n(n+1)/2. A positive divisor k of n is non-isolated if either k-1 or k+1 also divides n.

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A133996 a(n) = the smallest positive integer with exactly n positive "non-isolated divisors". A divisor, k, of n is non-isolated if (k-1) or (k+1) also 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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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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A328510 Smallest number whose divisors have n non-singleton runs.

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A328511 Number of non-singleton runs of divisors of 2n.

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Maple

Mathematica

A356733 Number of neighborless parts in the integer partition with Heinz number n.

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