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A086971
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Number of semiprime divisors of n.
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33
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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
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OFFSET
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1,12
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COMMENTS
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REFERENCES
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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.
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LINKS
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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]
Eric Weisstein's World of Mathematics, Semiprime.
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FORMULA
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G.f.: Sum_{k = p*q, p prime, q prime} x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 25 2017
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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)
(Haskell)
a086971 = sum . map a064911 . a027750_row
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CROSSREFS
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Cf. A001358, A064911, A001221, A000005, A000010, A004018, A007913, A056170, A079275, A001222, A220264 (least inverse).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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