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

1,3

Comments

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

Links

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

Eric Weisstein's World of Mathematics, Divisor Function.

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(PARI) 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)
(PARI) a(n)=sumdiv(n, d, (n%(d+1)!=0)); \\ Joerg Arndt, Jan 06 2015
(Haskell)
a137921 n = length $ filter (> 0) $
   map ((mod n) . (+ 1)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 23 2011
(Python)
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

Crossrefs

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.

Sequence in context: A006374 A193677 A281855 * A064876 A262689 A319816

Adjacent sequences: A137918 A137919 A137920 * A137922 A137923 A137924

Keyword

nonn,nice

Author

Reinhard Zumkeller, Feb 23 2008

Extensions

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

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

Status

approved