A157358 -id:A157358 - 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).

A157357 Products of 3 distinct triple-safe primes.

Original entry on oeis.org

777239, 1555559, 3112199, 4409399, 10635959, 12192599, 23348519, 23796743, 30612839, 47610023, 48628127, 55778519, 67454423, 91581239, 95286263, 97290047, 99883319, 102996599, 104812679, 135002663, 137841647, 148398599, 162707543, 170450999, 172007639, 186520823

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

			777239=23*47*719; 23, 47, and 719 are triple-safe prime numbers.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A157358, A001358, A005384, A005385, A006881, A007304, A066179, A111206, A157342, A157344, A157345, A157346, A157347, A157352, A157353, A157354, A157355, A157356

lst={};Do[If[Plus@@Last/@FactorInteger[n]==3,a=Length[First/@FactorInteger[n]];If[a==3,b=First/@FactorInteger[n];c=b[[1]];d=b[[2]];e=b[[3]];If[PrimeQ[cx=(c-1)/2]&&PrimeQ[cy=(cx-1)/2]&&PrimeQ[(cy-1)/2]&&PrimeQ[dx=(d-1)/2]&&PrimeQ[dy=(dx-1)/2]&&PrimeQ[(dy-1)/2]&&PrimeQ[ex=(e-1)/2]&&PrimeQ[ey=(ex-1)/2]&&PrimeQ[(ey-1)/2],AppendTo[lst,n]]]],{n,9!,11!}];lst

list(lim)=my(v=List(), P=select(p->isprime(p\2) && isprime(p\4) && isprime(p\8), primes([11, sqrtint(lim\11+1)-1])), p, q, t); for(i=1, #P, p=P[i]; if(p^3>=lim, break); for(j=i+1, #P, q=P[j]; t=p*q; forprime(r=q+4, lim\t, if(isprime(r\2) && isprime(r\4) && isprime(r\8), listput(v, r*t))))); Set(v); \\ Charles R Greathouse IV, Oct 14 2021

a(5)-a(26) from Charles R Greathouse IV, Oct 14 2021

A157359 Quatro-safe primes.

Original entry on oeis.org

47, 1439, 2879, 858239, 861599, 982559, 1014719, 1067999, 2029439, 2034239, 2297759, 2683679, 2978399, 3301919, 4068479, 4288799, 4737599, 5454719, 6484319, 6753119, 7145759, 8624159, 9511199, 9717119, 10533599, 10739999

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

(47-1)/2=23,(23-1)/2=11,(11-1)/2=5,(5-1)/2=2(prime),...

Cf. A005385, A066179, A157357, A157358

lst={};Do[p=Prime[n];If[PrimeQ[a=(p-1)/2]&&PrimeQ[b=(a-1)/2]&&PrimeQ[c=(b-1)/2]&&PrimeQ[(c-1)/2],AppendTo[lst,p]],{n,10!}];lst
Select[Prime[Range[711000]],AllTrue[Rest[NestList[(#-1)/2&,#,4]], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 31 2018 *)

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 *)

A328799 Primes p that are simultaneously of the forms 2q+1, 4r+3, 6s+5 and 8t+7 where q,r,s,t are primes.

Original entry on oeis.org

23, 47, 1439, 2879, 11279, 51599, 209519, 243119, 349199, 507359, 700319, 903359, 1190639, 1342079, 1650959, 1956719, 2978399, 3304079, 3376559, 3841679, 4858559, 5404319, 5454719, 6207599, 6486479, 7682399, 7825439, 8169599, 8826479, 8970959, 9546959

Offset: 1

J. M. Bergot and Robert Israel, Nov 18 2019

nonn

All terms == 23 (mod 24). All but the first == 47 (mod 48).

			a(3)=1439 is a term because 1439=2*719+1=4*359+3=6*239+5=8*179+7 and 1439, 719, 359, 239 and 179 are all primes.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A005385, A066179, A157358.

map(t -> 24*t+23, select(k -> andmap(isprime, [3*k+2,4*k+3,6*k+5,12*k+11,24*k+23]), [0, seq(k,k=1..10^6,2)]));

Clarified definition. - N. J. A. Sloane, Nov 15 2021

A162021 Triple-safe primes which are also triple-Sophie Germain primes.

Original entry on oeis.org

8981279, 17313839, 18635759, 82062479, 82479119, 98517599, 112242479, 113164319, 152799359, 184829279, 193409039, 230749199, 296709839, 305598719, 339116159, 393280799, 406283519

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

The intersection of the primes in A157358 and those in A023272: they remain prime after each
of three successive applications of the substitution p->(p-1)/2, and remain prime after each
three successive applications of the substitution p->2p+1. Therefore the sequence is a subsequence
of A162019.
They appear for example in the middle of chains started in A059767 or in even longer Cunningham chains. [R. J. Mathar, Jun 26 2009].

f[n_]:=Module[{x},If[PrimeQ[(n-1)/2]&&PrimeQ[(((n-1)/2)-1)/2]&&PrimeQ[(((((n-1)/ 2)-1)/2)-1)/2]&&PrimeQ[2*n+1]&&PrimeQ[2*(2*n+1)+1]&&PrimeQ[2*(2*(2*n+1)+1)+1], x=1,x=0];x]; lst={};Do[p=Prime[n];If[f[p]!=0,AppendTo[lst,p]],{n, 6*10!}];lst

Edited by R. J. Mathar, Jun 26 2009

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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A157357 Products of 3 distinct triple-safe primes.

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A157359 Quatro-safe primes.

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

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A328799 Primes p that are simultaneously of the forms 2*q+1, 4*r+3, 6*s+5 and 8*t+7 where q,r,s,t are primes.

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Examples

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A162021 Triple-safe primes which are also triple-Sophie Germain primes.

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Programs

Mathematica

Extensions

A328799 Primes p that are simultaneously of the forms 2q+1, 4r+3, 6s+5 and 8t+7 where q,r,s,t are primes.