A292596 -id:A292596 - cp's OEIS frontend

A292598 a(n) is the number of odd primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 2, 1, 2, 2, 2, 0, 1, 0, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3, 2, 3, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 3, 4, 2, 3, 3, 3, 0, 0, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 3, 3, 4, 4, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A010051, A065091, A078349, A292596.

a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + a(floor(n/2)).
a(n) = A000120(A292596(n)).
For all n >= 1, a(n) <= A078349(n).

A292599 a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 6, 7, 4, 5, 6, 6, 8, 9, 8, 9, 12, 12, 14, 15, 8, 8, 10, 10, 12, 13, 12, 13, 16, 16, 18, 18, 16, 17, 18, 18, 24, 25, 24, 25, 28, 28, 30, 31, 16, 16, 16, 16, 20, 21, 20, 20, 24, 24, 26, 27, 24, 25, 26, 26, 32, 32, 32, 33, 36, 36, 36, 37, 32, 33, 34, 34, 36, 36, 36, 37, 48, 48, 50, 51, 48, 48, 50, 50, 56, 57, 56, 56, 60, 60, 62, 62, 32

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

1-bits in base-2 expansion of a(n) indicate the positions of primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A010051, A078349, A292258, A292259, A292936.

Cf. also A292596 (variant for odd primes).

A292599 := proc(n)
    option remember;
    if n = 1 then
        0 ;
    else
        A010051(n) + 2*procname(floor(n/2)) ;
    end if;
end proc:
seq(A292599(n),n=1..100) ; # R. J. Mathar, Sep 28 2017

a[1] = 0; a[n_] := a[n] = Boole[PrimeQ[n]] + 2*a[Floor[n/2]]; Array[a, 96] (* Jean-François Alcover, Sep 29 2017 *)

a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).
Other identities. For all n >= 1:
A000120(a(n)) = A078349(n).
A007814(1+a(n)) = A292936(n).

A292597 a(1) = 1; for n > 1, a(n) = c(n) + 2*a(floor(n/2)), where c(n) is the characteristic function of odd composites, A071904.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 4, 8, 9, 8, 8, 8, 8, 8, 9, 16, 16, 18, 18, 16, 17, 16, 16, 16, 17, 16, 17, 16, 16, 18, 18, 32, 33, 32, 33, 36, 36, 36, 37, 32, 32, 34, 34, 32, 33, 32, 32, 32, 33, 34, 35, 32, 32, 34, 35, 32, 33, 32, 32, 36, 36, 36, 37, 64, 65, 66, 66, 64, 65, 66, 66, 72, 72, 72, 73, 72, 73, 74, 74, 64, 65, 64, 64, 68, 69, 68, 69, 64, 64, 66, 67

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

1-bits in base-2 expansion of a(n) indicate the positions of odd nonprimes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A000035, A010051, A014076, A071904, A292596, A292598.

a(1) = 1; for n > 1, a(n) = (A000035(n)*(1-A010051(n))) + 2*a(floor(n/2)).

For all n >= 1, a(n) + A292596(n) = n.

cp's OEIS Frontend

A292598 a(n) is the number of odd primes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

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A292599 a(1) = 0; for n > 1, a(n) = A010051(n) + 2*a(floor(n/2)).

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Formula

A292597 a(1) = 1; for n > 1, a(n) = c(n) + 2*a(floor(n/2)), where c(n) is the characteristic function of odd composites, A071904.

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