A292599 -id:A292599 - cp's OEIS frontend

A292936 a(n) = the least k >= 0 such that floor(n/(2^k)) is a nonprime; a(n) is degree of the "safeness" of prime, 0 if n is not a prime, 1 for unsafe primes (A059456), and k >= 2 for primes that are (k-1)-safe but not k-safe.

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

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

Records occur at positions 1, 2, 5, 11, 23, 47, 2879, ... (A292937).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007814, A078349, A292599, A292937.
Cf. A000040, A005385, A066179, A157358, A157359 (positions of terms that are > k, for k = 0..4).
Cf. A059456 (positions of ones).

A292936 := proc(n)
    for k from 0 do
        if not isprime(floor(n/2^k)) then
            return k;
        end if;
    end do:
end proc:
seq(A292936(n),n=1..100) ; # R. J. Mathar, Sep 28 2017

Table[SelectFirst[Range[0, 10], ! PrimeQ@ Floor[n/(2^#)] &], {n, 105}] (* Michael De Vlieger, Sep 29 2017 *)

A292936(n) = { my(k=0); while(isprime(n), n >>= 1; k++); k; };

(define (A292936 n) (A007814 (1+ (A292599 n))))

a(n) = A007814(1+A292599(n)).
For n >= 1, a(n) <= A078349(n).
For n > 47, a(n) <= A007814(1+n).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A292936 a(n) = the least k >= 0 such that floor(n/(2^k)) is a nonprime; a(n) is degree of the "safeness" of prime, 0 if n is not a prime, 1 for unsafe primes (A059456), and k >= 2 for primes that are (k-1)-safe but not k-safe.

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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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