A112967 -id:A112967 - cp's OEIS frontend

A112968 a(n) = Sum_{i+j=n} mu(i)*Omega(j), with mu=A008683 and Omega=A001222.

0, 0, 1, 0, 0, -2, -2, -2, -2, -2, -6, -2, -4, -2, -7, -1, -5, 0, -7, -3, -9, 1, -11, 2, -7, 1, -12, 1, -11, 7, -8, -5, -8, -1, -18, 3, -10, 1, -13, 1, -7, 13, -12, -2, -13, 6, -16, 3, -11, 3, -15, -4, -16, 13, -15, -4, -15, 4, -17, 11, -14, 4, -13, 7, -12, 15, -17, -5, -15, 16, -13, 3, -12, 3, -20, 3, -27, 19, -20, -3, -11, 3

Offset: 1

Reinhard Zumkeller, Oct 07 2005

			a(5) = mu(1)*Omega(4)+mu(2)*Omega(3)+mu(3)*Omega(2)+mu(4)*Omega(1) = 1*2 - 1*1 - 1*1 + 0*1 = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A013939, A112967, A068341, A112962, A112963, A112964, A112966.

a112968 n = sum $ zipWith (*)
   a008683_list $ reverse $ take (n - 1) a001222_list
-- Reinhard Zumkeller, Feb 29 2012

A112968[n_]:=Plus@@Table[MoebiusMu[i]*PrimeOmega[n-i],{i,1,n-1}]; Array[A112968,200] (* Enrique Pérez Herrero, Feb 28 2012 *)

Corrected by N. J. A. Sloane, Mar 01 2006

A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 8, 0, 2, 2, 10, 0, 8, 0, 8, 2, 2, 0, 20, 1, 2, 4, 8, 0, 12, 0, 20, 2, 2, 2, 24, 0, 2, 2, 20, 0, 12, 0, 8, 8, 2, 0, 40, 1, 8, 2, 8, 0, 20, 2, 20, 2, 2, 0, 34, 0, 2, 8, 35, 2, 12, 0, 8, 2, 12, 0, 52, 0, 2, 8, 8, 2, 12, 0, 40, 10, 2, 0, 34, 2, 2, 2, 20, 0, 34, 2, 8, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Aug 24 2018

nonn

Dirichlet convolution of A001222 with itself.

			24 has 8 divisors, namely 1, 2, 3, 4, 6, 8, 12, 24, and four prime factors counted with multiplicity. The divisors have 0, 1, 1, 2, 2, 3, 3, 4 divisors respectively. So a(24) = 0 * (4 - 0) + 1 * (4 - 1) + 1 * (4 - 1) + 2 * (4 - 2) + 2 * (4 - 2) + 3 * (4 - 3) + 4 * (4 - 4) = 0 + 3 + 3 + 4 + 4 + 3 + 3 + 0 = 20. - _David A. Corneth_, Jan 12 2019

Antti Karttunen, Table of n, a(n) for n = 1..20160
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A008578 (positions of 0's), A069264, A070288, A112967, A317938, A322375.

f:= proc(n) local F,G,t,x;
   F:= map(t -> t[2], ifactors(n)[2]);
   G:= unapply(normal(mul((1-x^(t+1))/(1-x), t = F)),x);
  (convert(F,`+`)-1)*D(G)(1) - (D@@2)(G)(1);
end proc:
map(f, [$1..100]); # Robert Israel, Jan 17 2019

Table[Sum[PrimeOmega[d] PrimeOmega[n/d], {d, Divisors[n]}], {n, 95}]

a(n) = sumdiv(n, d, bigomega(d)*bigomega(n/d)); \\ Michel Marcus, Aug 25 2018

a(n) = bn = bigomega(n); sumdiv(n, d, bd = bigomega(d); bd * (bn - bd)) \\ David A. Corneth, Jan 12 2019

a(A025487(n)) = A322375(n). - David A. Corneth, Jan 12 2019
From Robert Israel, Jan 17 2019: (Start)
If x and y are coprime, a(x*y) = a(x)*A000005(y) + A000005(x)*a(y) + A000005(x*y)*A001222(x)*A001222(y).
If p is prime, a(p^k) = (k^3-k)/6 = A000292(k-1). (End)

A112965 a(n) = Sum_{i+j=n} omega(i)*omega(j), where omega = A001221.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 7, 8, 9, 10, 14, 14, 17, 18, 23, 24, 27, 26, 32, 32, 35, 36, 44, 42, 47, 48, 52, 50, 58, 54, 65, 62, 67, 66, 78, 70, 79, 78, 88, 84, 94, 88, 100, 100, 103, 100, 118, 106, 119, 114, 124, 116, 135, 122, 138, 134, 141, 136, 155, 142, 155, 154, 163, 156

Offset: 1

Reinhard Zumkeller, Oct 07 2005

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A013939, A112966, A112967.

Table[Sum[PrimeNu[i]*PrimeNu[n - i], {i, n - 1}], {n, 65}] (* Ivan Neretin, Jan 21 2017 *)

a(n) = sum(i=2, n-2, omega(i)*omega(n-i)); \\ Michel Marcus, Jan 22 2017

G.f.: (Sum_{k>=1} x^prime(k)/(1 - x^prime(k)))^2. - Ilya Gutkovskiy, Jan 31 2017

A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 10, 23, 32, 64, 98, 187, 296, 543, 891, 1595, 2660, 4694, 7924, 13854, 23556, 40940, 69939, 121122, 207490, 358517, 615292, 1061635, 1824013, 3144404, 5406257, 9314645, 16021922, 27595176, 47478950, 81757104, 140691461, 242232918, 416890645, 717712748, 1235289624

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001222.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A001222, A022559, A112967, A129921, A180305, A293549, A300671.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[bigomega](i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 41; CoefficientList[Series[1/(1 - Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[1/(1 - Sum[PrimeOmega[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeOmega[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 41}]

G.f.: 1/(1 - Sum_{k>=2} A001222(k)*x^k).

cp's OEIS Frontend

A112968 a(n) = Sum_{i+j=n} mu(i)*Omega(j), with mu=A008683 and Omega=A001222.

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A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

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A112965 a(n) = Sum_{i+j=n} omega(i)*omega(j), where omega = A001221.

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A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

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