A181064 -id:A181064 - cp's OEIS frontend

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

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 *)

A181062 Prime powers minus 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 22, 24, 26, 28, 30, 31, 36, 40, 42, 46, 48, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 88, 96, 100, 102, 106, 108, 112, 120, 124, 126, 127, 130, 136, 138, 148, 150, 156, 162, 166, 168, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226

Offset: 1

Matthew Vandermast, Oct 07 2010

If 0 is excluded, a(n) gives the possible lengths of the longest string of consecutive divisors of a positive integer: range of values of A055874.
a(n) is the largest number m such that A051451(n) = A003418(m).
From Jianing Song, Nov 01 2023: (Start)
Let q = A000961(n) for n > 1. Then:
 - a(n) is the number of units in the finite field F_q.
 - a(n) is the number of solutions to x*y = t for any t != 0 in F_q.
 - If q is odd, then a(n) is also the number of solutions to x^2 - y^2 = t for any t != 0 in F_q. (End)

			Any integer that is divisible by 5 consecutive integers will be divisible by at least 6 consecutive integers. Hence 5 is not in the sequence.

Matthew House, Table of n, a(n) for n = 1..10000

Cf. A000961, A181063, A181064, A181121.
Includes A006093 as a subsequence.
Cf. A003418, A051451, A055874.

Join[{0}, Select[Range@225, PrimePowerQ] - 1] (* Ivan Neretin, Aug 04 2016 *)

isA181062(n)=ispower(n++,,&n);isprime(n) \\ Charles R Greathouse IV, Jun 29 2011

isok(n) = (n==0) || isprimepower(n++); \\ Michel Marcus, Aug 05 2016

a(n) = A000961(n)-1.

Entry revised by N. J. A. Sloane, Jan 06 2013

cp's OEIS Frontend

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

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