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A086971 Number of semiprime divisors of n. 15
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

Smallest n with exactly k (k=0..12) semiprime divisors are: 1, 4, 12, 30, 60, 180, 210, 420, 1260, 6300, 2310, 4620, 13860. See A220264. - Zak Seidov, Mar 31 2011

Also the number of congruences of sp (mod n) == k for which only one member exists, with sp being all the semiprimes and k<n, as it turns out, will be the semiprime divisors of n. As an example, let n=12, and the set of all the semiprimes, sp, (lets just use those sp < 10^6), only two congruences have only one member and they are 4 & 6. The congruence, sp (mod 12) == 1 has 35045 members. Mod[sp, 12] = {{0, 0}, {1, 35045}, {2, 20733}, {3, 14286}, {4, 1}, {5, 34984}, {6, 1}, {7, 34872}, {9, 14378}, {10, 20803}, {11, 34932}}. These are the same divisors of 12 found to be semiprimes. If only the other hand, some particular congruency, (mod j) with only one member is == 0, then j is a semiprime and therefore none of its 4 divisors will be a semiprime. Thus a(j)=1. - Robert G. Wilson v, Dec 10 2012

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)). - Reinhard Zumkeller, Dec 14 2012

REFERENCES

Hardy, G. H. and Wright, E. M. 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, 789-803, 1975.

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 arXiv version]

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

If n = p1^e1 * p2^e2 * ... * pj^ej for primes p1, p2, ..., pj and integer exponents e1, e2, ..., ej, then a(n) = |{k: ek >=2}| + T(j-1) where T(k) is the k-th triangular number A000217(k). The proof follows from the observation that any prime factor is either the square of a prime if that prime squared is a factor of n, or the product of 2 distinct primes in the factorization of n, which is the binomial coefficient C(j, 2) = T(j-1). - Jonathan Vos Post, Dec 08 2004

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

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

a(n) = sum(A064911(A027750(n,k)): k=1..A000005(n)). - Reinhard Zumkeller, Dec 14 2012

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 *)

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

Cf. A001358, A064911, A001221, A000005, A000010, A004018, A007913, A056170, A079275, A001222, A220264 (least inverse).

Sequence in context: A084114 A294881 A110475 * A211159 A088434 A205745

Adjacent sequences:  A086968 A086969 A086970 * A086972 A086973 A086974

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Sep 22 2003

EXTENSIONS

Entry revised by N. J. A. Sloane, Mar 28 2006

STATUS

approved

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Last modified December 14 12:04 EST 2019. Contains 329979 sequences. (Running on oeis4.)