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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 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); m:=nops(l);        m*(m-1)/2 +add(`if`(i>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); 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.)