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
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
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).
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