A086971 - OEIS

A086971

Number of semiprime divisors of n.

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

Inverse Moebius transform of A064911. - Jonathan Vos Post, Dec 08 2004

REFERENCES

G. H. Hardy and E. M. Wright, Section 17.10 in An Introduction to the Theory of Numbers, 5th ed., Oxford, England: Clarendon Press, 1979.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

E. A. Bender and J. R. Goldman, On the Applications of Mobius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, (1975), 789-803.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

N. J. A. Sloane, Transforms.

Eric Weisstein's World of Mathematics, Semiprime.

Eric Weisstein's World of Mathematics, Divisor Function.

Eric Weisstein's World of Mathematics, Moebius Transform.

FORMULA

a(n) = A106404(n) + A106405(n). - Reinhard Zumkeller, May 02 2005

a(n) = omega(n/core(n)) + binomial(omega(n),2) = A001221(n/A007913(n)) + binomial(A001221(n),2) = A056170(n) + A079275(n). - Rick L. Shepherd, Mar 06 2006

From Reinhard Zumkeller, Dec 14 2012: (Start)

a(n) = Sum_{k=1..A000005(n)} A064911(A027750(n,k)).

a(A220264(n)) = n and a(m) <> n for m < A220264(n); a(A008578(n)) = 0; a(A002808(n)) > 0; for n > 1: a(A102466(n)) <= 1 and a(A102467(n)) > 1; A066247(n) = A057427(a(n)). (End)

G.f.: Sum_{k = p*q, p prime, q prime} x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 25 2017

MAPLE

a:= proc(n) local l, m; l:=ifactors(n)[2]; m:=nops(l);

m*(m-1)/2 +add(`if`(i[2]>1, 1, 0), i=l)

end:

seq(a(n), n=1..120); # Alois P. Heinz, Jul 18 2013

MATHEMATICA

semiPrimeQ[n_] := PrimeOmega@ n == 2; f[n_] := Length@ Select[Divisors@ n, semiPrimeQ@# &]; Array[f, 105] (* Zak Seidov, Mar 31 2011 and modified by Robert G. Wilson v, Dec 08 2012 *)

a[n_] := Count[e = FactorInteger[n][[;; , 2]], _?(# > 1 &)] + (o = Length[e])*(o - 1)/2; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

PROG

(PARI) /* The following definitions of a(n) are equivalent. */

a(n) = sumdiv(n, d, bigomega(d)==2)

a(n) = f=factor(n); j=matsize(f)[1]; sum(m=1, j, f[m, 2]>=2) + binomial(j, 2)

a(n) = f=factor(n); j=omega(n); sum(m=1, j, f[m, 2]>=2) + binomial(j, 2)

a(n) = omega(n/core(n)) + binomial(omega(n), 2)

/* Rick L. Shepherd, Mar 06 2006 */

(Haskell)

a086971 = sum . map a064911 . a027750_row

-- Reinhard Zumkeller, Dec 14 2012

CROSSREFS