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

Author

Reinhard Zumkeller, Feb 23 2008

Keywords

• nonn
• nice

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

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor Function.

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.

Programs

Haskell

a137921 n = length $ filter (> 0) $
   map ((mod n) . (+ 1)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 23 2011

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 *)

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

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

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

Extensions

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

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