A161840 - OEIS

%I #37 Sep 22 2024 14:54:50

%S 0,0,0,2,0,2,0,2,2,2,0,4,0,2,2,4,0,4,0,4,2,2,0,6,2,2,2,4,0,6,0,4,2,2,

%T 2,8,0,2,2,6,0,6,0,4,4,2,0,8,2,4,2,4,0,6,2,6,2,2,0,10,0,2,4,6,2,6,0,4,

%U 2,6,0,10,0,2,4,4,2,6,0,8,4,2,0,10,2,2,2,6,0,10,2,4,2,2,2,10,0,4,4,8

%N Number of noncentral divisors of n.

%C Noncentral divisors in the following sense: if we sort the divisors of n in natural order, there is one "central", median divisor if the number of divisors tau(n) = A000005(n) is odd, and there are two "central" divisors if tau(n) is even. a(n) is the number of divisors not counting the median or two central divisors.

%H Antti Karttunen, <a href="/A161840/b161840.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F a(n) = tau(n)-2 + (tau(n) mod 2), tau = A000005.

%F a(n) = A000005(n) - A049240(n) - 1.

%F a(n) = A000005(n) + A010052(n) - 2.

%F a(n) = A000005(n) - A169695(n).

%F For n >= 2, a(n) = A200213(n) + 2*A010052(n). - _Antti Karttunen_, Jul 07 2017

%F a(n) = 2*A072670(n-1). - _Omar E. Pol_, Jul 08 2017

%F Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Jan 14 2024

%e The divisors of 4 are 1, 2, 4 so the noncentral divisors of 4 are 1, 4 because its central divisor is 2.

%e The divisors of 12 are 1, 2, 3, 4, 6, 12 so the noncentral divisors of 12 are 1, 2, 6, 12 because its central divisors  are 3, 4.

%p A000005 := proc(n) numtheory[tau](n) ; end: A010052 := proc(n) if issqr(n) then 1; else 0 ; fi; end: A161840 := proc(n) A000005(n)+A010052(n)-2 ; end: seq(A161840(n),n=1..100) ; # _R. J. Mathar_, Jul 04 2009

%t If[EvenQ[#],#-2,#-1]&/@DivisorSigma[0,Range[100]] (* _Harvey P. Dale_, Sep 22 2024 *)

%o (PARI) A161840(n) = numdiv(n)+issquare(n)-2; \\ _Antti Karttunen_, Jul 07 2017

%o (Scheme) (define (A161840 n) (+ (A000005 n) (A010052 n) -2)) ;; _Antti Karttunen_, Jul 07 2017

%Y Cf. A000005, A001620, A049240, A010052, A161841, A169695, A183002, A183003, A200213, A323643.

%K easy,nonn

%O 1,4

%A _Omar E. Pol_, Jun 21 2009

%E More terms from _R. J. Mathar_, Jul 04 2009