A328448 -id:A328448 - 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

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

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.

A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 17 2019

nonn

By convention, a(1) = 1, and a(p) = 0 for p prime.

			The non-singleton runs of the nontrivial divisors of 1260 are: {2,3,4,5,6,7} {9,10} {14,15} {20,21} {35,36}, so a(1260) = 6.

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

Positions of first appearances are A328459.
Positions of 0's and 1's are A088723.
The version that looks at all divisors 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).
Cf. A033676, A060681, A060775, A070824, A088725, A129308, A181063, A199970, A328165, A163870, A328194, A328448, A328449.

Table[Switch[n,1,1,?PrimeQ,0,,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,100}]

A328458(n) = if(1==n,n,my(rl=0,pd=0,m=0); fordiv(n, d, if(1(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

A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

Original entry on oeis.org

1, 2, 6, 12, 60, 420, 504, 840, 2520, 27720, 360360, 720720, 4084080

Offset: 0

Gus Wiseman, Oct 17 2019

			The sequence of terms > 1 together with their nontrivial divisors begins:
    2: {}
    6: {2,3}
   12: {2,3,4,6}
   60: {2,3,4,5,6,10,12,15,20,30}
  420: {2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210}
  504: {2,3,4,6,7,8,9,12,14,18,21,24,28,36,42,56,63,72,84,126,168,252}

Positions of first appearances in A328458.
The version for all divisors is A051451.
Cf. A000005, A033676, A060775, A070824, A119313, A129308, A181063, A199970, A163870, A328448, A328449.

dav=Table[Switch[n,1,1,_,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,1000}];
Table[Position[dav,i][[1,1]],{i,Union[dav]}]//Sort

a(12) from Robert Israel, Mar 31 2023

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A137921 Number of divisors d of n such that d+1 is not a divisor 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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A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

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A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n.

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A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

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