A106404 -id:A106404

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A086971

Number of semiprime divisors of n.

+10
33

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

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

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Sep 22 2003

EXTENSIONS

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

STATUS

approved

A305614

Expansion of Sum_{p prime} x^p/(1 + x^p).

+10
14

0, 0, 1, 1, -1, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 2, -1, 1, 0, 1, -2, 2, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 2, 0, 2, -2, 1, 0, 2, -2, 1, -1, 1, -2, 2, 0, 1, -2, 1, 0, 2, -2, 1, 0, 2, -2, 2, 0, 1, -3, 1, 0, 2, -1, 2, -1, 1, -2, 2, -1, 1, -2, 1, 0, 2, -2, 2, -1

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

OFFSET

0,13

COMMENTS

a(n) is the number of prime divisors p|n such that n/p is odd, minus the number of prime divisors p|n such that n/p is even.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = -Sum_{p|n prime} (-1)^(n/p).

From Robert Israel, Jun 07 2018: (Start)

If n is odd, a(n) = A001221(n).

If n == 2 (mod 4), a(n) = 2 - A001221(n).

If n == 0 (mod 4) and n > 0, a(n) = -A001221(n). (End)

L.g.f.: log(Product_{k>=1} (1 + x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018

EXAMPLE

The prime divisors of 12 are 2, 3. We see that 12/2 = 6, 12/3 = 4. None of those are odd, but both of them are even, so a(12) = -2.

The prime divisors of 30 are {2,3,5} with quotients {15,10,6}. One of these is odd and two are even, so a(30) = 1 - 2 = -1.

MAPLE

a:= n-> -add((-1)^(n/i[1]), i=ifactors(n)[2]):

seq(a(n), n=0..100); # Alois P. Heinz, Jun 07 2018

# Alternative

N:= 1000: # to get a(0)..a(N)

V:= Vector(N):

p:= 1:

p:= nextprime(p);

if p > N then break fi;

R:= [seq(i, i=p..N, p)];

W:= <seq((-1)^(n+1), n=1..nops(R))>;

V[R]:= V[R]+W;

od:

[0, seq(V[i], i=1..N)]; # Robert Israel, Jun 07 2018

MATHEMATICA

Table[Sum[If[PrimeQ[d], (-1)^(n/d - 1), 0], {d, Divisors[n]}], {n, 30}]

CROSSREFS

Cf. A000005, A000607, A001221, A008683, A048165, A083399, A088705, A106404, A205745, A280954.

KEYWORD

sign

AUTHOR

Gus Wiseman, Jun 06 2018

STATUS

approved

A205745

a(n) = card { d | d*p = n, d odd, p prime }

+10
5

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2

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

OFFSET

1,15

COMMENTS

Equivalently, a(n) is the number of prime divisors p|n such that n/p is odd. - Gus Wiseman, Jun 06 2018

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

FORMULA

O.g.f.: Sum_{p prime} x^p/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018

Sum_{k=1..n} a(k) = (n/2) * (log(log(n)) + B) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

MATHEMATICA

a[n_] := Sum[ Boole[ OddQ[d] && PrimeQ[n/d] ], {d, Divisors[n]} ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 27 2013 *)

PROG

(Sage)

def A205745(n):

return sum((n//d) % 2 for d in divisors(n) if is_prime(d))

[A205745(n) for n in (1..105)]

(PARI) a(n)=if(n%2, omega(n), n%4/2) \\ Charles R Greathouse IV, Jan 30 2012

(Haskell)

a205745 n = sum $ map ((`mod` 2) . (n `div`))

[p | p <- takeWhile (<= n) a000040_list, n `mod` p == 0]

-- Reinhard Zumkeller, Jan 31 2012

CROSSREFS

Cf. A000005, A000607, A001221, A008683, A010051, A068050, A077761, A083399, A088705, A106404, A305614.

KEYWORD

nonn

AUTHOR

Peter Luschny, Jan 30 2012

STATUS

approved

A106405

Number of odd semiprimes dividing n.

+10
3

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

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

OFFSET

1,45

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A086971(n) - A106404(n);

a(A046315(n)) = 1; a(A093641(n)) = 0; a(A105441(n)) > 0.

EXAMPLE

a(105) = #{15, 21, 35} = #{3*5, 3*7, 5*7} = 3.

MATHEMATICA

Table[Count[Divisors[n], _?(OddQ[#]&&PrimeOmega[#]==2&)], {n, 120}] (* Harvey P. Dale, May 05 2015 *)

a[n_] := Count[e = FactorInteger[n][[;; , 2]], _?(# > 1 &)] + (o = Length[e])*(o - 1)/2 - If[EvenQ[n], If[e[[1]] > 1, o, o - 1], 0]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

CROSSREFS

Cf. A000005, A001227, A046315, A086971, A093641, A105441, A106404.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 02 2005

STATUS

approved

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