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%I #20 Mar 22 2024 06:49:09
%S 0,2,0,2,0,3,0,2,0,2,0,4,0,2,0,2,0,3,0,4,0,2,0,4,0,2,0,2,0,5,0,2,0,2,
%T 0,4,0,2,0,4,0,5,0,2,0,2,0,4,0,2,0,2,0,3,0,4,0,2,0,6,0,2,0,2,0,3,0,2,
%U 0,2,0,6,0,2,0,2,0,3,0,4,0,2,0,6,0,2,0,2,0,7,0,2,0,2,0,4,0,2,0,4,0,3,0,2,0
%N a(n) = number of non-isolated divisors of n.
%C A divisor d of n is non-isolated if either d-1 or d+1 divides n. a(2n-1) = 0 for all n >= 1.
%H Ray Chandler, <a href="/A132747/b132747.txt">Table of n, a(n) for n=1..10000</a>
%F a(n) = A000005(n) - A132881(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2) + 1 = A002162 + 1 = 1.693147.... . - _Amiram Eldar_, Mar 22 2024
%e The positive divisors of 20 are 1,2,4,5,10,20. Of these, 1 and 2 are next to each other and 4 and 5 are next to each other. So a(20) = the number of these divisors, which is 4.
%t Table[Length[Select[Divisors[n], If[ # > 1, IntegerQ[n/(#*(# - 1))]] || IntegerQ[n/(#*(# + 1))] &]], {n, 1, 90}] (* _Stefan Steinerberger_, Oct 26 2007 *)
%o (PARI) a(n) = my(div = divisors(n)); sumdiv(n, d, vecsearch(div, d-1) || vecsearch(div, d+1)); \\ _Michel Marcus_, Aug 22 2014
%Y Cf. A000005, A002162, A129308, A132748, A132881.
%K nonn
%O 1,2
%A _Leroy Quet_, Aug 27 2007
%E More terms from _Stefan Steinerberger_, Oct 26 2007
%E Extended by _Ray Chandler_, Jun 24 2008
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