A055874 -id:A055874 - cp's OEIS frontend

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

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.

A235918 Largest m such that 1, 2, ..., m divide n^2.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1

Offset: 1

Michel Marcus, Jan 17 2014

nonn

Note that a(n) is equal to A071222(n-1) = A053669(n)-1 for the first 209 values of n. The first difference occurs at n=210, where a(210)=7, while A071222(209)=10. A235921 lists all n where a(n) differs from A071222(n-1). (Note also that a(n) is equal to A071222(n+29) for n=1..179.) - [Comment revised by Antti Karttunen, Jan 26 2014 because of the changed definition of A235921 and newly inserted a(0)=1 term of A071222.]
See A055874 for a similar comment concerning the difference between A055874 and A232098.
Average value is 1.9124064... = sum_{n>=1} 1/A019554(A003418(n)). - Charles R Greathouse IV, Jan 24 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

One less than A236454.

Cf. also A055874, A053669, A055926, A071222, A232098, A235921.

a[n_] := Module[{m = 1}, While[Divisible[n^2, m++]]; m - 2]; Array[a, 100] (* Jean-François Alcover, Mar 07 2016 *)

a(n) = my(m = 1); while ((n^2 % m) == 0, m++); m - 1; \\ Michel Marcus, Jan 17 2014

a(n) = A055874(n^2).

a(n) = A236454(n)-1.

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&]&]

A328450 Numbers that are a smallest number with k pairs of successive divisors, for some k.

Original entry on oeis.org

1, 2, 6, 12, 60, 72, 180, 360, 420, 840, 1260, 2520, 3780, 5040, 13860, 27720, 36960, 41580, 55440, 83160, 166320, 277200, 360360, 471240, 491400, 720720, 1081080, 1113840, 2162160, 2827440, 3341520, 4324320, 5405400, 6126120

Offset: 0

Gus Wiseman, Oct 15 2019

nonn

A sorted version of A287142.

			The divisors of 72 are {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, with pairs of successive divisors {{1, 2}, {2, 3}, {3, 4}, {8, 9}}, and no smaller number has 4 successive pairs, so 72 belongs to the sequence.

Sorted positions of first appearances in A129308.
The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a longest run of length n is A328449(n).
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A072627, A181063, A199970, A287142, A328165.

dat=Table[Count[Differences[Divisors[n]],1],{n,10000}];
Sort[Table[Position[dat,i][[1,1]],{i,Union[dat]}]]

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

A344181 Numbers such that repeated division by their largest factorial divisor (as long as such a divisor larger than one exists) eventually yields 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 96, 120, 128, 144, 192, 240, 256, 288, 384, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1440, 1536, 1920, 2048, 2304, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5760, 6144, 6912, 7680, 8192, 8640, 9216, 10080

Offset: 1

Antti Karttunen, May 18 2021

nonn

Numbers whose closure under map x -> A076934(x) contains 1.
Largest factorial divisor of n is A000142(A055874(n)).
These numbers could be called "greedy Jordan-Polya numbers", as their presence in A001013 can be determined by a simple greedy algorithm.

			12 = 2^2 * 3 is present, as the largest factorial that divides 12 is A000142(3) = 6, and then 12/6 = 2 is also divisible by a factorial, and then 2/A000142(2) = 1.

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial numbers

Cf. A000142, A055874, A076934.
Positions of ones in A093411.
Subsequence of A001013. A344179 gives the terms not present here.
Cf. also A025487 (analogous sequence for primorials).

fctdiv[n_] := Module[{m = 1, k = 1}, While[Divisible[n, m], k++; m *= k]; m /= k; n/m]; Select[Range[10^4], FixedPoint[fctdiv, #] == 1 &] (* Amiram Eldar, May 22 2021 *)

A055770 Largest factorial number which divides n.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6

Offset: 1

Labos Elemer, Jul 12 2000

Largest m! which divides n.

			3! = 6 divides 12, so a(12) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.

Cf. A000142, A055881 (values of the m's), A055926, A055874, A073575.

Cf. also A053589.

With[{rf=Reverse[Range[7]!]},Table[SelectFirst[rf,Divisible[n,#]&],{n,120}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 05 2017 *)

A055770(n) = { my(m=1, i=2); while(!(n%m), m *= i; i++); return(m/(i-1)); } \\ Antti Karttunen, Dec 19 2018

a(n) = A000142(A055881(n)). - Antti Karttunen, Dec 19 2018

Name changed, old name moved to comments by Antti Karttunen, Dec 19 2018

A126800 Smallest divisor of n which is greater than largest divisor d of n such that each integer from 1 to d is also a divisor of n.

Original entry on oeis.org

3, 4, 5, 6, 7, 4, 3, 5, 11, 6, 13, 7, 3, 4, 17, 6, 19, 4, 3, 11, 23, 6, 5, 13, 3, 4, 29, 5, 31, 4, 3, 17, 5, 6, 37, 19, 3, 4, 41, 6, 43, 4, 3, 23, 47, 6, 7, 5, 3, 4, 53, 6, 5, 4, 3, 29, 59, 10, 61, 31, 3, 4, 5, 6, 67, 4, 3, 5, 71, 6, 73, 37, 3, 4, 7, 6, 79, 4

Offset: 3

Leroy Quet, Feb 21 2007

nonn

a(n) is the smallest divisor of n which is greater than A055874(n).

a(n) is also the smallest divisor m, m > 1, of n where m - 1 is not a divisor of n.

			The divisors of 12 are 1, 2, 3, 4, 6, 12. The first four divisors are the first four positive integers, but 5 is not a divisor of 12, and the smallest divisor greater than 5 is 6, so a(12) = 6.
The divisors of 14 are 1, 2, 7, 14. The first two divisors are the first two positive integers, but 3 is not a divisor of 14, and the smallest divisor greater than 3 is 7, so a(14) = 7.

Michael De Vlieger, Table of n, a(n) for n = 3..10000

Cf. A055874.

A055874 := proc(n) local m; for m from 1 do if n mod m <> 0 then RETURN(m-1) ; fi ; od: end: A126800 := proc(n) local a; for a from A055874(n)+1 do if n mod a = 0 then RETURN(a) ; fi ; od: end: seq(A126800(n),n=3..80) ; # R. J. Mathar, Nov 01 2007

sdn[n_]:=Module[{divs=Divisors[n],s,c},s=First[Split[Differences[divs]]];c=Length[s]+1;Which[PrimeQ[n],n,First[s]>1,divs[[2]],True,First[Drop[ divs,c]]]];Array[sdn,80,3] (* Harvey P. Dale, Jan 18 2015 *)
Array[#[[1 + LengthWhile[Prepend[Differences@ #, 1], # == 1 &] ]] &@ Divisors@ # &, 78, 3] (* Michael De Vlieger, Oct 10 2017 *)

a(n) = n if n is an odd prime or if n = 4 or 6. - Alonso del Arte, Aug 07 2014

More terms from R. J. Mathar, Nov 01 2007

A195155 Number of divisors d of n such that d-1 also divides n or d-1 = 0.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 2

Offset: 1

Omar E. Pol, Sep 19 2011

First differs from A055874 at a(20).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A007862, A049820, A055874, A129308, A137921, A195150, A195153.

with(numtheory):
a:= n-> add(`if`(d=1 or irem(n, d-1)=0, 1, 0), d=divisors(n)):
seq(a(n), n=1..200);  # Alois P. Heinz, Oct 17 2011

d1[n_]:=Module[{d=Rest[Divisors[n]]},Count[d,?(Divisible[n,#-1]&)]+1]; Array[d1, 90] (* _Harvey P. Dale, Oct 31 2013 *)

from itertools import pairwise
from sympy import divisors
def A195155(n): return 1 if n&1 else 1+sum(1 for a, b in pairwise(divisors(n)) if a+1==b) # Chai Wah Wu, Jun 09 2025

a(n) = A000005(n) - A195150(n).
a(n) = 1 + A129308(n).
a(2n-1) = 1; a(2n) = 1 + A007862(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Dec 31 2023
a(n) <= A038548(n) <= A000005(n). - Charles R Greathouse IV, Jun 09 2025

A344179 Jordan-Polya numbers (A001013) not in A344181.

Original entry on oeis.org

72, 216, 432, 1296, 1728, 2592, 5184, 7776, 10368, 14400, 15552, 28800, 31104, 41472, 46656, 51840, 57600, 62208, 93312, 115200, 120960, 124416, 155520, 186624, 230400, 248832, 279936, 311040, 373248, 460800, 559872, 604800, 746496, 921600, 933120, 995328, 1088640, 1119744, 1209600, 1244160, 1492992, 1679616, 1728000

Offset: 1

Antti Karttunen, May 18 2021

nonn

These are numbers that are products of factorial numbers (A000142), but whose presence in A001013 cannot be determined by a simple greedy algorithm that repeatedly divides the largest factorial divisor [= A055874(n)!] off, until only 1 remains.

			72 = 2*6*6 = 2! * 3! * 3! is present in A001013, and as it is not present in A344181 (because when it is divided by its largest factorial divisor 24, we get 72/24 = 3, an odd number that is not a factorial itself), it is therefore present in this sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial numbers

Setwise difference of A001013 and A344181.

Cf. A000142, A055874, A076934, A093411.

fct = Array[#! &, 10]; prev = {}; jp = fct; While[jp != prev, prev = jp; jp = Select[Union @@ Outer[Times, jp, fct], # <= fct[[-1]] &]]; fctdiv[n_] := Module[{m = 1, k = 1}, While[Divisible[n, m], k++; m *= k]; m /= k; n/m]; Select[jp, FixedPoint[fctdiv, #] != 1 &] (* Amiram Eldar, May 22 2021 *)

search_up_to = 2^22;
A076934(n) = for(k=2, oo , if(n%k, return(n), n /= k));
A093411(n) = if(!n,n, if(n%2, n, A093411(A076934(n))));
A001013list(lim, mx=lim)=if(lim<2, return([1])); my(v=[1], t=1); for(n=2, mx, t*=n; if(t>lim, break); v=concat(v, t*A001013list(lim\t, t))); Set(v) \\ From A001013
v001013 = A001013list(search_up_to);
A001013(n) = v001013[n];
isA344179(n) = if(v001013[#v001013]A093411(n))&&vecsearch(v001013,n)));
for(n=1,search_up_to,if(isA344179(n),print1(n,", ")));

cp's OEIS Frontend

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

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A235918 Largest m such that 1, 2, ..., m divide n^2.

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

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A328450 Numbers that are a smallest number with k pairs of successive divisors, for some k.

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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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A344181 Numbers such that repeated division by their largest factorial divisor (as long as such a divisor larger than one exists) eventually yields 1.

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A055770 Largest factorial number which divides n.

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A126800 Smallest divisor of n which is greater than largest divisor d of n such that each integer from 1 to d is also a divisor of n.

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