A292598 -id:A292598 - cp's OEIS frontend

A078349 Number of primes in sequence h(m) defined by h(1) = n, h(m+1) = Floor(h(m)/2).

0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 2, 2, 2, 3, 2, 3, 1, 1, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 5, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 3, 4, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 2, 3, 3, 3, 4, 3, 3, 4, 4, 5, 5, 1, 2, 1, 1, 1

Offset: 1

Joseph L. Pe, Dec 23 2002

nonn

			The sequence h(m) for n = 5 is 5, 2, 1, 0, 0, 0, ...., in which two terms are primes. Therefore a(5) = 2.

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

Cf. A010051, A292258, A292259, A292598, A292599, A292936.

f[n_] := Module[{i, p}, i = n; p = 0; While[i > 1, If[PrimeQ[i], p = p + 1]; i = Floor[i/2]]; p]; Table[f[i], {i, 1, 100}]

A078349(n) = if(1==n,0,isprime(n)+A078349(n\2)); \\ Antti Karttunen, Oct 01 2017

From Antti Karttunen, Oct 01 2017: (Start)
a(1) = 0; for n > 1, a(n) = A010051(n) + a(floor(n/2)).
a(n) = A000120(A292599(n)).
a(n) = A007814(A292258(n)).
a(n) >= A292598(n).
a(n) >= A292936(n).
(End)

A292596 a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + 2*a(floor(n/2)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 0, 2, 3, 4, 5, 6, 6, 0, 1, 0, 1, 4, 4, 6, 7, 8, 8, 10, 10, 12, 13, 12, 13, 0, 0, 2, 2, 0, 1, 2, 2, 8, 9, 8, 9, 12, 12, 14, 15, 16, 16, 16, 16, 20, 21, 20, 20, 24, 24, 26, 27, 24, 25, 26, 26, 0, 0, 0, 1, 4, 4, 4, 5, 0, 1, 2, 2, 4, 4, 4, 5, 16, 16, 18, 19, 16, 16, 18, 18, 24, 25, 24, 24, 28, 28, 30, 30, 32, 33, 32, 32, 32, 33

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

1-bits in base-2 expansion of a(n) indicate the positions of odd 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, A292597, A292598.

Cf. also A292599 (variant for all primes).

a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + 2*a(floor(n/2)).
Other identities. For all n >= 1:
a(n) + A292597(n) = n.
A000120(a(n)) = A292598(n).
A007814(1+a(n)) <= A007814(1+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

A078349 Number of primes in sequence h(m) defined by h(1) = n, h(m+1) = Floor(h(m)/2).

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A292596 a(1) = a(2) = 0; for n > 2, a(n) = A010051(n) + 2*a(floor(n/2)).

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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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