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A137921 Number of divisors d of n such that d+1 is not a divisor of n. 8
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 (list; graph; refs; listen; history; text; internal format)
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

a(n) <= A000005(n), with equality iff n is odd; a(A137922(n)) = 2.

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) - A129308(n). - Michel Marcus, Jan 06 2015

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-Fran├žois 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. Cf. A000005.

Sequence in context: A006374 A193677 A281855 * A064876 A262689 A105517

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

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Last modified November 23 21:46 EST 2017. Contains 295141 sequences.