A157359 -id:A157359 - 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).

A162019 Double-safe primes which are also double-Sophie Germain primes.

Original entry on oeis.org

11, 359, 719, 214559, 215399, 245639, 253679, 266999, 507359, 508559, 574439, 670919, 744599, 825479, 1017119, 1072199, 1184399, 1363679, 1621079, 1688279, 1786439, 2156039, 2377799, 2429279, 2633399, 2684999, 2900039, 3103799

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 24 2009

nonn

The intersection of the primes in A066179 and those in A007700: they remain prime after each of two successive applications of the substitution p->(p-1)/2, and remain prime after each two successive applications of the substitution p->2p+1.

			a(1)=11 is double safe: (11-1)/2=5; (5-1)/2=2, and double Sophie-Germain: 2*11+1=23; 2*23+1=47.

Cf. A007700, A023302, A066179, A157359.

lst={};Do[p=Prime[n];If[PrimeQ[safe=(p-1)/2],If[PrimeQ[(safe-1)/2],If[PrimeQ[sophie=2*p+1],If[PrimeQ[2*sophie+1],AppendTo[lst,p]]]]],{n,3*9!}];lst

a(n) = 4*A023302(n) + 3 = (A157359(n)-3)/4. - R. J. Mathar, Jun 26 2009

Edited by R. J. Mathar, Jun 26 2009

A292937 a(0)=1, followed by highly safe primes: positions of records in A292936.

Original entry on oeis.org

1, 2, 5, 11, 23, 47, 2879, 71850239, 2444789759, 21981381119

Offset: 0

Antti Karttunen, Sep 28 2017

The starting offset is 0 to accommodate 1, which is only nonprime in this sequence, and also to align with the indexing used in A110056.

Sequence starts like A007505, and at least for terms a(5) .. a(9) is equal to A110056.

Cf. A007505, A066174, A110056, A292936.

Cf. A000040, A005385, A066179, A157358, A157359 (each starts with the term a(1) .. a(5) of this sequence).

With[{s = Table[SelectFirst[Range[0, 10], ! PrimeQ@ Floor[n/(2^#)] &], {n, 10^7}]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Sep 29 2017 *)

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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A162019 Double-safe primes which are also double-Sophie Germain primes.

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A292937 a(0)=1, followed by highly safe primes: positions of records in A292936.

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