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A048105 Number of non-unitary divisors of n. 42
0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 0, 2, 5, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 8, 0, 2, 2, 5, 0, 0, 0, 4, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Number of zeros in row n of table A225817; a(n) = A000005(n) - A034444(n); for n > 1: a(n) = A000005(n) - 2 * A007875(n). - Reinhard Zumkeller, Jul 30 2013

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Sigma(0, n) - 2^r(n), where r()=A001221, the number of distinct primes dividing n.

Dirichlet g.f.: zeta(s)^2 - zeta(s)^2/zeta(2*s). - Geoffrey Critzer, Dec 10 2014

G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 21 2017

EXAMPLE

Example 1: If n is squarefree (A005117) then a(n)=0 since all divisors are unitary.

Example 2: n=12, d(n)=6, ud(n)=4, nud(12)=d(12)-ud(12)=2; from {1,2,3,4,6,12} {1,3,4,12} are unitary while {2,6} are not unitary divisors.

Example 3: n=p^k, a true prime power, d(n)=k+1, u(d)=2^r(x)=2, so nud(n)=d(p^k)-2=k+1 i.e., it can be arbitrarily large.

MATHEMATICA

Table[DivisorSigma[0, n] - 2^PrimeNu[n], {n, 1, 50}] (* Geoffrey Critzer, Dec 10 2014 *)

PROG

(Haskell)

a048105 n = length [d | d <- [1..n], mod n d == 0, gcd d (n `div` d) > 1]

-- Reinhard Zumkeller, Aug 17 2011

(PARI) a(n)=my(f=factor(n)[, 2]); prod(i=1, #f, f[i]+1)-2^#f \\ Charles R Greathouse IV, Sep 18 2015

CROSSREFS

a(n) = A000005(n) - A034444(n).

Cf. A056170.

Sequence in context: A054673 A155103 A295819 * A335021 A176202 A040081

Adjacent sequences:  A048102 A048103 A048104 * A048106 A048107 A048108

KEYWORD

nonn

AUTHOR

Labos Elemer

STATUS

approved

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Last modified September 23 09:03 EDT 2020. Contains 337298 sequences. (Running on oeis4.)