A292936 -id:A292936 - cp's OEIS frontend

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

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 *)

A156659 Characteristic function of safe primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Feb 13 2009

nonn

R. Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Safe prime
Index entries for characteristic functions

Cf. A005385, A156660, A292936.

a156659 n = fromEnum $ a010051 n == 1 && a010051 (n `div` 2) == 1
-- Reinhard Zumkeller, Sep 18 2011

Array[Boole[And[PrimeQ@ #, PrimeQ[(# - 1)/2]]] &, 105, 0] (* Michael De Vlieger, Dec 16 2017 *)

a(n) = isprime(n) && isprime(floor((n-1)/2)) \\ Iain Fox, Dec 17 2017

a(n) = if n and also (n-1)/2 is prime then 1 else 0;
a(A005385(n)) = 1; a(A156657(n)) = 0; a(A059456(n)) = 0.
a(n) = A010051(n)*A010051((n-1)/2).
A156875(n) = Sum_{k=1..n} a(k). - Reinhard Zumkeller, Feb 18 2009
a(n) = 1 iff A292936(n) > 1. - Antti Karttunen, Dec 15 2017

A063377 Sophie Germain degree of n: number of iterations of n under f(k) = 2k+1 before we reach a number that is not a prime.

Original entry on oeis.org

0, 5, 2, 0, 4, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 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, 6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Reiner Martin, Jul 14 2001

nonn

a(n) >= 1 means that n is prime; a(n) >= 2 means that n is a Sophie Germain prime. Is the Sophie Germain degree always finite? Is it unbounded?
A339579 is an essentially identical sequence from 1981. - N. J. A. Sloane, Dec 24 2020
From Michael S. Branicky, Dec 24 2020: (Start)
All n > 5 with a(n) >= 4 satisfy n == 9 (mod 10).
Proof.  Let f^k(n) denote iterates of 2*k + 1, with f^0(n) = n.
n != 0, 2, 4, 5, 6, or 8 (mod 10), otherwise f^0(n) is not prime, and a(n) = 0.
n != 7 (mod 10) otherwise f^1(n) = 2*n + 1 == 5 (mod 10), not prime, and a(n) <= 1.
n != 3 (mod 10) otherwise f^2(n) = 4*r + 3 == 5 (mod 10), not prime, and a(n) <= 2.
n != 1 (mod 10) otherwise f^3(n) = 8*r + 7 == 5 (mod 10), not prime, and a(n) <= 3.
(End)
From Peter Schorn, Jan 18 2021: (Start)
The Sophie Germain degree is always finite.
Proof. Let f^k(n) denote iterates of 2*k + 1 with closed form f^k(n) = 2^k * n + 2^k - 1.
There are three cases for n:
1. If n is not a prime then f^0(n) = n is composite.
2. If n = 2 then f^5(2) = 95 is composite.
3. If n is an odd prime then f^(n-1)(n) = 2^(n-1) * n + 2^(n-1) - 1 is divisible by n since 2^(n-1) == 1 (mod n) by Fermat's theorem.
(End)

			a(2)=5 because 2, 5, 11, 23, 47 are prime but 95 is not.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A005384, A063378, A093008, A093007, A292936, A339579.
For records see A339581.
See also Cunningham chains, A005602, A005603.

Table[Length[NestWhileList[2#+1&,n,PrimeQ[#]&]],{n,100}]-1 (* Harvey P. Dale, Aug 08 2020 *)

a(n) = {if (! isprime(n), return (0)); d = 1; k = n; while(isprime(p = 2*k+1), k = p; d++;); return (d);} \\ Michel Marcus, Jul 22 2013

From Michael S. Branicky, Dec 24 2020: (Start)
See proof above.
a(n) = 0 if n == 0, 2, 4, 5, 6, 8 (mod 10), and n != 2 or 5.
a(n) <= 1 if n == 7 (mod 10).
a(n) <= 2 if n == 3 (mod 10).
a(n) <= 3 if n == 1 (mod 10).
(End)

Term a(1) = 0 prepended by Antti Karttunen, Oct 09 2018.

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)

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).

A323161 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=3) = -n, f(n) = 0 if n-1 is an odd prime, and f(n) = floor((n-1)/2) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 4, 7, 7, 8, 4, 9, 4, 10, 10, 11, 4, 12, 4, 13, 13, 14, 4, 15, 15, 16, 16, 17, 4, 18, 4, 19, 19, 20, 20, 21, 4, 22, 22, 23, 4, 24, 4, 25, 25, 26, 4, 27, 27, 28, 28, 29, 4, 30, 30, 31, 31, 32, 4, 33, 4, 34, 34, 35, 35, 36, 4, 37, 37, 38, 4, 39, 4, 40, 40, 41, 41, 42, 4, 43, 43, 44, 4, 45, 45, 46, 46, 47, 4, 48, 48, 49, 49, 50, 50, 51, 4, 52, 52, 53, 4, 54, 4

Offset: 1

Antti Karttunen, Jan 06 2019

nonn

For all i, j: A322809(i) = A322809(j) <=> a(i+1) = a(j+1).
For all i, j: a(i) = a(j) => b(i) = b(j), where b can be, but is not limited to, any of the following sequences: A029834, A049084, A062590, A063377, A064891, A078442 (A049076), A175663, A175682, A269668, A292936, A323162, many of which are related to counting primes in certain kinds of chains or iterations.
Why does this work? Consider the function f given in the definition: based on its properties, we can deduce from the value of f(n) the following information about n:
  (A) If f(n) = -2, then n is 2, the only even prime,
  (B) If f(n) = -3, then n is 3, the first odd prime,
  (C) If f(n) is zero, then n is an even composite preceded by a prime, but we don't know which even composite exactly,
  (D) If f(n) > 0 and f(1+2*f(n)) = f(2+2*f(n)), then n is either (D1) an odd composite number, or (D2) an even composite number preceded by an odd composite number, and the said composite number in both cases is 1 + 2*f(n),
  (E) If f(n) > 0 and f(1+2*f(n)) <> f(2+2*f(n)), then n is an odd prime > 3, specifically, 1 + 2*f(n).
As this sequence is a restricted growth sequence transform of the said function f, we have a(i) = a(j) <=> f(i) = f(j) for all i, j, thus, even without knowing the value of n, but just a(n), we can find the value of f(n) by searching for the minimal k such that a(k) = a(n), then compute f(k) with that k. Furthermore, any function g defined as g(n) = h(f(n)) [where h is any function], clearly satisfies
  a(i) = a(j) => g(i) = g(j), for all i, j. [*]
For instances of such functions g, we can consider many sequences like those sequences b(n) listed above, that have g(n) = 0 for all composite numbers, and g(p) > 0 for all primes p. This is usually the pattern, but there are exceptions, like A323162, which is the characteristic function of A005381, composites n such that n-1 is also composite. These are precisely the numbers that occur twice in this sequence, while all other numbers (including primes), occur just once, that is, reside in their own singular equivalence classes. Thus, it is not guaranteed that all sequences g matching to this sequence (i.e. those satisfying the implication *), even if not false positives in strict sense, would necessarily have some consistent relation to primes, instead, they might contain any random values at the positions given by A093515. However, in the current OEIS, such sequences are exceedingly rare.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A305801, A305900, A322809.
Cf. A005381 (numbers that occur twice in this sequence), A093515 (numbers > 1 that occur just once).
Cf. A010051, A029834, A049076, A049084, A062590, A063377, A064891, A078442, A175663, A175682, A269668, A292936, A323162 (some of the matched sequences).

up_to = 10000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A323161aux(n) = if(n<=3,-n,if(isprime(n-1),0,((n-1)>>1))); \\ This implements the function f of the definition.
v323161 = rgs_transform(vector(up_to,n,A323161aux(n)));
A323161(n) = v323161[n];

a(1) = 1; for n > 1, a(n) = 1 + A322809(n-1).

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

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A156659 Characteristic function of safe primes.

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A063377 Sophie Germain degree of n: number of iterations of n under f(k) = 2k+1 before we reach a number that is not a prime.

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

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A323161 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=3) = -n, f(n) = 0 if n-1 is an odd prime, and f(n) = floor((n-1)/2) for all other numbers.

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