A161840 - OEIS

A161840

Number of noncentral divisors of n.

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, 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, 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

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OFFSET

1,4

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

FORMULA

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

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

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

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

For n >= 2, a(n) = A200213(n) + 2*A010052(n). - Antti Karttunen, Jul 07 2017

a(n) = 2*A072670(n-1). - Omar E. Pol, Jul 08 2017

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

EXAMPLE

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

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.

MAPLE

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

MATHEMATICA

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

PROG

(PARI) A161840(n) = numdiv(n)+issquare(n)-2; \\ Antti Karttunen, Jul 07 2017

(Scheme) (define (A161840 n) (+ (A000005 n) (A010052 n) -2)) ;; Antti Karttunen, Jul 07 2017

CROSSREFS

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

Sequence in context: A246721 A249441 A076472 * A355851 A140302 A085341

Adjacent sequences: A161837 A161838 A161839 * A161841 A161842 A161843

KEYWORD

easy,nonn

AUTHOR

Omar E. Pol, Jun 21 2009

EXTENSIONS

More terms from R. J. Mathar, Jul 04 2009

STATUS

approved