A199970 -id:A199970 - cp's OEIS frontend

A137921 Number of divisors d of n such that d+1 is not a divisor of n.

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

Offset: 1

Reinhard Zumkeller, Feb 23 2008

a(n) = number of "divisor islands" of n. A divisor island is any set of consecutive divisors of a number where no pairs of consecutive divisors in the set are separated by 2 or more. - Leroy Quet, Feb 07 2010

			The divisors of 30 are 1,2,3,5,6,10,15,30. The divisor islands are (1,2,3), (5,6), (10), (15), (30). (Note that the differences between consecutive divisors 5-3, 10-6, 15-10 and 30-15 are all > 1.) There are 5 such islands, so a(30)=5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.

Bisections: A099774, A174199.
First appearance of n is at position A173569(n).
Numbers whose divisors have no non-singleton runs are A005408.
The longest run of divisors of n has length A055874(n).
The number of successive pairs of divisors of n is A129308(n).
Cf. A000005, A001620, A027750, A060680, A088723, A088725, A181063, A199970, A328165, A328166, A328448, A328450.

a137921 n = length $ filter (> 0) $
   map ((mod n) . (+ 1)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 23 2011

with(numtheory): disl := proc (b) local ct, j: ct := 1: for j to nops(b)-1 do if 2 <= b[j+1]-b[j] then ct := ct+1 else end if end do: ct end proc: seq(disl(divisors(n)), n = 1 .. 120); # Emeric Deutsch, Feb 12 2010

f[n_] := Length@ Split[ Divisors@n, #2 - #1 == 1 &]; Array[f, 105] (* f(n) from Bobby R. Treat *) (* Robert G. Wilson v, Feb 22 2010 *)
Table[Count[Differences[Divisors[n]],?(#>1&)]+1,{n,110}] (* _Harvey P. Dale, Jun 05 2012 *)
a[n_] := DivisorSum[n, Boole[!Divisible[n, #+1]]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=my(d,s=0);if(n%2,numdiv(n),d=divisors(n);for(i=1,#d,if(n%(d[i]+1),s++));s)

a(n)=sumdiv(n,d,(n%(d+1)!=0)); \\ Joerg Arndt, Jan 06 2015

from sympy import divisors
def A137921(n):
    return len([d for d in divisors(n,generator=True) if n % (d+1)])
# Chai Wah Wu, Jan 05 2015

a(n) <= A000005(n), with equality iff n is odd; a(A137922(n)) = 2.
a(n) = A000005(n) - A129308(n). - Michel Marcus, Jan 06 2015
a(n) = A001222(A328166(n)). - Gus Wiseman, Oct 16 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

Corrected and edited by Charles R Greathouse IV, Apr 19 2010

Edited by N. J. A. Sloane, Aug 10 2010

A129308 a(n) is the number of positive integers k such that k*(k+1) divides n.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 1, 0

Offset: 1

Leroy Quet, May 26 2007

nonn

The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.

In other words, a(n) is the number of oblong numbers (A002378) dividing n. - Bernard Schott, Jul 29 2022

			The divisors of 20 are 1,2,4,5,10,20. Of these there are two that are of the form k(k+1): 2 = 1*2 and 20 = 4*5. So a(2) = 2.

Ray Chandler, Table of n, a(n) for n = 1..10000
P. Erdős and R. R. Hall, On some unconventional problems on the divisors of integers, J. Austral. Math. Soc., Ser. A, 25, 479-485 (1978).
MathOverflow, On the number of consecutive divisors of an integer.

Positions of 0's and 1's are A088725, whose characteristic function is A360128.
First appearance of n is A287142(n), with sorted version A328450.
The longest run of divisors of n has length A055874(n).
One less than A195155.
Cf. A000005, A002378, A003601, A007862, A027750, A033676, A060680, A060681, A072627, A088722, A181063, A199970, A328026, A328165, A328166.
Cf. A005369, A059841, A287142, A344005.

a = {}; For[n = 1, n < 90, n++, k = 1; co = 0; While[k < Sqrt[n], If[IntegerQ[ n/(k*(k + 1))], co++ ]; k++ ]; AppendTo[a, co]]; a (* Stefan Steinerberger, May 27 2007 *)
Table[Count[Differences[Divisors[n]],1],{n,30}] (* Gus Wiseman, Oct 15 2019 *)

a(n)=sumdiv(n, d, n%(d+1)==0); \\ Michel Marcus, Jan 06 2015

from itertools import pairwise
from sympy import divisors
def A129308(n): return 0 if n&1 else sum(1 for a, b in pairwise(divisors(n)) if a+1==b) # Chai Wah Wu, Jun 09 2025

a(2n-1) = 0; a(2n) = A007862(n). - Ray Chandler, Jun 24 2008
G.f.: Sum_{n>=1} x^(n*(n+1))/(1-x^(n*(n+1))). - Joerg Arndt, Jan 30 2011 [modified by Ilya Gutkovskiy, Apr 14 2021]
a(n) = A000005(n) - A137921(n), where A137921(n) is the number of maximal runs of successive divisors of n. - Gus Wiseman, Oct 15 2019
a(n) = Sum_{d|n} A005369(d). - Ridouane Oudra, Jan 22 2021
a(n) = A195155(n)-1. - Antti Karttunen, Feb 21 2023
From Amiram Eldar, Dec 31 2023: (Start)
a(n) = A088722(n) + A059841(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)

More terms from Stefan Steinerberger, May 27 2007

Extended by Ray Chandler, Jun 24 2008

A328166 Heinz number of the run-lengths of the divisors of n.

Original entry on oeis.org

2, 3, 4, 6, 4, 10, 4, 12, 8, 12, 4, 28, 4, 12, 16, 24, 4, 40, 4, 36, 16, 12, 4, 112, 8, 12, 16, 48, 4, 120, 4, 48, 16, 12, 16, 224, 4, 12, 16, 144, 4, 120, 4, 48, 64, 12, 4, 448, 8, 48, 16, 48, 4, 160, 16, 144, 16, 12, 4, 832, 4, 12, 64, 96, 16, 160, 4, 48, 16

Offset: 1

Gus Wiseman, Oct 07 2019

nonn

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

			Splitting the divisors of 30 into runs gives {{1, 2, 3}, {5, 6}, {10}, {15}, {30}}, and the Heinz number of {1, 1, 1, 2, 3} is 120, so a(30) = 120.
More examples from _Antti Karttunen_, Dec 09 2021: (Start)
Splitting the divisors of 1 into runs gives {1}, and the Heinz number of that is 2.
Splitting the divisors of 2 into runs gives {1, 2}, and the Heinz number of that is 3. [one run of length 2, therefore a(2) = prime(2)^1].
Splitting the divisors of 3 into runs gives {1} and {3}, and the Heinz number of that is 4. [two runs of length 1, therefore a(3) = prime(1)^2].
Splitting the divisors of 4 into runs gives {1, 2} and {4}, and the Heinz number of that is 6. [one run of length 1, and other run of length 2, therefore a(4) = prime(1)*prime(2)].
Splitting the divisors of 5 into runs gives {1} and {5}, and the Heinz number of that is 4. [two runs of length 1, therefore a(5) = prime(1)^2].
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Wikipedia, Run (cards)
Index entries for sequences related to Heinz numbers

The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the set of divisors of n is A275700(n).
Numbers whose divisors do not have weakly decreasing run-lengths are A328165.
Cf. A000005, A056239, A060680, A060775, A119313, A137921, A181063, A199970.

Table[Times@@Prime/@Length/@Split[Divisors[n],#2==#1+1&],{n,30}]

A328166(n) = { my(rl=0,pd=0,v=vector(numdiv(n)),m=1); fordiv(n, d, if(d>(1+pd), v[rl]++; rl=0); pd=d; rl++); v[rl]++; for(i=1,#v, m *= prime(i)^v[i]); (m); }; \\ Antti Karttunen, Dec 09 2021

A001222(a(n)) = A137921(n).

A056239(a(n)) = A000005(n).

A181063 Smallest positive integer with a discrete string of exactly n consecutive divisors, or 0 if no such integer exists.

Original entry on oeis.org

1, 2, 6, 12, 3960, 60, 420, 840, 17907120, 2520, 411863760, 27720, 68502634200, 447069823200, 360360, 720720, 7600186994400, 12252240, 9524356075634400, 81909462250455840, 1149071006394511200, 232792560, 35621201198229847200, 5354228880, 91351145008363640400

Offset: 1

Matthew Vandermast, Oct 07 2010

nonn

The word "discrete" is used to describe a string of consecutive divisors that is not part of a longer such string.
Does a(n) ever equal 0?
a(n) = A003418(n) iff n belongs to A181062; otherwise, a(n) > A003418(n). a(A181062(n)) = A051451(n).

			a(5) = 3960 is divisible by 8, 9, 10, 11, and 12, but not 7 or 13. It is the smallest positive integer with a string of 5 consecutive divisors that is not part of a longer string.
From _Gus Wiseman_, Oct 16 2019: (Start)
The sequence of terms together with their divisors begins:
     1: {1}
     2: {1,2}
     6: {1,2,3,6}
    12: {1,2,3,4,6,12}
  3960: {1,2,...,8,9,10,11,12,...,1980,3960}
    60: {1,2,3,4,5,6,...,30,60}
   420: {1,2,3,4,5,6,7,...,210,420}
   840: {1,2,3,4,5,6,7,8,...,420,840}
(End)

David W. Wilson, Table of n, a(n) for n = 1..1000

The version taking only the longest run is A328449.
The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
Cf. A000005, A003418, A027750, A060775, A181064, A199970, A328166, A328448.

tav=Table[Length/@Split[Divisors[n],#2==#1+1&],{n,10000}];
Table[Position[tav,i][[1,1]],{i,Split[Union@@tav,#2==#1+1&][[1]]}] (* Assumes there are no zeros. - Gus Wiseman, Oct 16 2019 *)

A328457 Length of the longest run of divisors > 1 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Records occur at A328448.
Positions of 0's and 1's are A088725.
The version that looks at all divisors (including 1) is A055874.
The number of successive pairs of divisors > 1 of n is A088722(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The longest run of nontrivial divisors of n is A328458(n).
Cf. A000005, A027750, A129308, A181063, A199970, A328162, A328195, A328449, A360128.

Table[If[n==1,0,Max@@Length/@Split[Rest[Divisors[n]],#2==#1+1&]],{n,100}]

A328457(n) = { my(rl=0,pd=0,m=0); fordiv(n, d, if(d>1, if(d>(1+pd), m = max(m,rl); rl=0); pd=d; rl++)); max(m,rl); }; \\ Antti Karttunen, Feb 23 2023

Data section extended up to a(105) by Antti Karttunen, Feb 23 2023

A199969 a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.

Original entry on oeis.org

0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Jaroslav Krizek, Nov 26 2011

From Paul Curtz, Feb 09 2015: (Start)
The nonnegative numbers with 0 instead of 1. See A254667(n), which is linked to the Bernoulli numbers A164555(n)/A027642(n), an autosequence of the second kind.
Offset 0 could be chosen.
An autosequence of the second kind is a sequence whose main diagonal is the first upper diagonal multiplied by 2. If the first upper diagonal is
s0, s1, s2, s3, s4, s5, ...,
the sequence is
Ssk(n) = 2*s0, s0, s0 + 2*s1, s0 +3*s1, s0 + 4*s1 + 2*s2, s1 + 5*s1 + 5*s2, etc.
The corresponding coefficients are A034807(n), a companion to A011973(n).
The binomial transform of Ssk(n) is (-1)^n*Ssk(n).
Difference table of a(n):
   0,  0, 2, 3, 4, 5, 6, 7, ...
   0,  2, 1, 1, 1, 1, 1, ...
   2, -1, 0, 0, 0, 0 ...
  -3,  1, 0, 0, 0, ...
   4, -1, 0, 0, ...
  -5,  1, 0, ...
   6, -1, ...
   7, ...
  etc.
a(n) is an autosequence of the second kind. See A054977(n).
The corresponding autosequence of the first kind (a companion) is 0, 0 followed by the nonnegative numbers (A001477(n)). Not in the OEIS.
Ssk(n) = 2*Sfk(n+1) - Sfk(n) where Sfk(n) is the corresponding sequence of the first kind (see A254667(n)).
(End)
Number of binary sequences of length n-1 that contain exactly one 0 and at least one 1. - Enrique Navarrete, May 11 2021

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A199968 (the smallest non-divisor h of n (1A199970. A001477, A011973, A034807, A054977, A254667.
Cf. A007978.
Essentially the same as A000027, A028310, A087156 etc.

Join[{0,0},Table[Max[Complement[Range[n],Divisors[n]]],{n,3,70}]] (* or *) Join[{0,0},Range[2,70]] (* Harvey P. Dale, May 31 2014 *)

if(n>2,n-1,0) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = n-1 for n >= 3.

E.g.f.: 1-x^2/2+(x-1)*exp(x). - Enrique Navarrete, May 11 2021

A328448 Smallest number whose divisors > 1 have a longest run of length n, and 0 if none exists.

Original entry on oeis.org

2, 6, 12, 504, 60, 420, 840, 4084080, 2520, 21162960, 27720, 2059318800, 0, 360360, 720720, 8494326640800, 12252240, 281206918792800, 0, 0, 232792560, 409547311252279200, 5354228880, 619808900849199341280, 26771144400, 54749786241679275146400, 80313433200, 5663770990518545704800

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

			The runs of divisors of 504 (greater than 1) are {{2,3,4},{6,7,8,9},{12},{14},{18},{21},{24},{28},{36},{42},{56},{63},{72},{84},{126},{168},{252},{504}}, the longest of which has length 4, and 504 is the smallest number with this property, so a(4) = 504.

The version that looks at all divisors (including 1) is A328449.
The longest run of divisors of n greater than 1 has length A328457.
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A000005, A027750, A055874, A060680, A181063, A199970, A328450.

Data corrected and extended by Giovanni Resta, Oct 18 2019

A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

Original entry on oeis.org

0, 1, 2, 6, 12, 0, 60, 420, 840, 0, 2520, 0, 27720, 0, 0, 360360, 720720, 0, 12252240, 0, 0, 0, 232792560, 0, 5354228880, 0, 26771144400, 0, 80313433200, 0, 2329089562800, 72201776446800, 0, 0, 0, 0, 144403552893600, 0, 0, 0, 5342931457063200, 0

Offset: 0

Gus Wiseman, Oct 16 2019

nonn

Wikipedia, Run (cards)

Positions of 0's are 0 followed by A024619 - 1.
The version that looks only at all divisors > 1 is A328448.
The longest run of divisors of n has length A055874.
The longest run of divisors of n greater than one has length A328457.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a (not necessarily longest) maximal run of length n is A181063.
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A199970, A328195.

tav=Table[Max@@Length/@Split[Divisors[n],#2==#1+1&],{n,10000}];
Table[If[FreeQ[tav,i],0,Position[tav,i][[1,1]]],{i,0,Max@@tav}]

a(n) = LCM(1,2,...,n) = A003418(n) if n + 1 is a prime power, otherwise a(n) = 0.

A328165 Numbers whose divisors do not have weakly decreasing run-lengths.

Original entry on oeis.org

56, 72, 110, 112, 132, 144, 156, 182, 210, 216, 224, 240, 264, 272, 288, 306, 312, 342, 364, 380, 392, 396, 420, 432, 440, 448, 462, 468, 480, 506, 528, 544, 550, 552, 576, 600, 612, 616, 624, 648, 650, 684, 702, 720, 728, 756, 760, 770, 780, 784, 792, 812

Offset: 1

Gus Wiseman, Oct 07 2019

nonn

			The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, with runs {{1, 2}, {4}, {7, 8}, {14}, {28}, {56}}, with lengths (2, 1, 2, 1, 1, 1), which are not weakly decreasing, so 56 is in the sequence.

Wikipedia, Run (cards)

The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A000005, A027750, A033676, A060680, A060775, A119313, A181063, A199970.

Select[Range[1000],!GreaterEqual@@Length/@Split[Divisors[#],#2==#1+1&]&]

A199968 a(n) = the smallest non-divisor h of n (1

Original entry on oeis.org

0, 0, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2

Offset: 1

Jaroslav Krizek, Nov 26 2011

nonn

a(n) = A173540(n,1). - Reinhard Zumkeller, Oct 02 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A199969 (the greatest non-divisor h of n (1A199970.

Cf. A173540.

a199968 = head . a173540_row  -- Reinhard Zumkeller, Oct 02 2015

a(n) = A007978(n) for n>=3.

cp's OEIS Frontend

A137921 Number of divisors d of n such that d+1 is not a divisor of n.

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A129308 a(n) is the number of positive integers k such that k*(k+1) divides n.

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A328166 Heinz number of the run-lengths of the divisors of n.

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A181063 Smallest positive integer with a discrete string of exactly n consecutive divisors, or 0 if no such integer exists.

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A328457 Length of the longest run of divisors > 1 of n.

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A199969 a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.

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A328448 Smallest number whose divisors > 1 have a longest run of length n, and 0 if none exists.

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