A033299 -id:A033299 - cp's OEIS frontend

A059323 Smaller term of closest safe prime pairs.

11, 47, 167, 347, 467, 1307, 2027, 2447, 4127, 4787, 5087, 5387, 5927, 12527, 12647, 15287, 18947, 28307, 39107, 39827, 41507, 44687, 51827, 63587, 64007, 71987, 73847, 76367, 76907, 78467, 79967, 83207, 118787, 121547, 143687, 164987

Offset: 1

Labos Elemer, Jan 26 2001

nonn

			11 and 23 are consecutive safe primes but not consecutive primes; 467 and 479 are consecutive safe primes and consecutive primes as well. Both 11 and 467 are here.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005383, A033299.

[p:p in PrimesUpTo(165000)| IsPrime(p+12) and IsPrime((p-1) div 2) and IsPrime( (p+11) div 2)]; // Marius A. Burtea, Jan 13 2020

seqQ[n_] := And @@ PrimeQ[{n, n+12, (n-1)/2, (n+11)/2}]; Select[Range[165000], seqQ] (* Amiram Eldar, Jan 13 2020 *)

{x| both x and x+12 are safe primes}; Intersection(12+A005385, primes)

Offset corrected by Amiram Eldar, Jan 13 2020

A059327 a(n) is smallest safe prime (A005385) such that a(n) + 12n is the next safe prime, i.e., x = (a(n) - 1)/2 and x + 6n are closest Sophie Germain primes.

Original entry on oeis.org

11, 23, 227, 179, 107, 1367, 263, 887, 2099, 719, 587, 8819, 3467, 1019, 10163, 27827, 1619, 7823, 27299, 2207, 44267, 3203, 7247, 5099, 11807, 45887, 18119, 15803, 79559, 13163, 40127, 42839, 20663, 79979, 17483, 53267, 47963, 33863

Offset: 1

Labos Elemer, Jan 26 2001

nonn

			{11, 23, 227, 179, 107, ...} are the smallest safe primes such that {11+12, 23+24, 227+36, 179+48, 107+60, ...} = {23, 47, 263, 227, 167, ...} are their next safe primes to which the corresponding Sophie Germain primes are {11, 23, 131, 113, 83, ...} respectively.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A033299.

safeQ[p_] := PrimeQ[(p-1)/2]; max = 38; seq = Table[0, {max}]; c=0; p1 = p2 = 11; While[c < max, p2 = NextPrime[p2]; If[safeQ[p2], d = (p2 - p1)/12; If[d <= max && seq[[d]] == 0, c++; seq[[d]] = p1]; p1 = p2]]; seq (* Amiram Eldar, Jan 13 2020 *)

Offset corrected by Amiram Eldar, Jan 13 2020

A059322 First differences of sequence of consecutive safe primes.

Original entry on oeis.org

2, 4, 12, 24, 12, 24, 24, 60, 12, 48, 36, 84, 12, 24, 84, 12, 24, 60, 24, 132, 120, 24, 24, 96, 36, 168, 96, 24, 12, 48, 72, 48, 36, 96, 204, 84, 120, 12, 24, 36, 108, 240, 12, 120, 240, 60, 24, 60, 36, 24, 96, 48, 36, 264, 156, 156, 24, 60, 84, 60, 72, 48, 12, 120, 24

Offset: 1

Labos Elemer, Jan 26 2001

nonn

Except for (5,7) and (7,11), all terms are divisible by 12, since safe primes are congruent to 5 modulo 6 except 7 and safe_prime + 6 is not a safe prime. Closest safe primes differ by 12 like (11,23) or (83207,83219).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5001 from G. C. Greubel)

Cf. A005385, A033299.

Differences[Select[Prime[Range[500]],PrimeQ[(#-1)/2]&]]  (* Harvey P. Dale, Jan 12 2011 *)

list(lim) = {my(p1 = 5); forprime(p2 = 7, lim, if(isprime((p2-1)/2), print1(p2-p1, ", "); p1 = p2));} \\ Amiram Eldar, Mar 02 2025

a(n) = A005385(n+1) - A005385(n). [corrected by Harvey P. Dale, Jan 12 2011 and Zak Seidov, Sep 19 2016]

Offset corrected by Amiram Eldar, Mar 02 2025

cp's OEIS Frontend

A059323 Smaller term of closest safe prime pairs.

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A059327 a(n) is smallest safe prime (A005385) such that a(n) + 12n is the next safe prime, i.e., x = (a(n) - 1)/2 and x + 6n are closest Sophie Germain primes.

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A059322 First differences of sequence of consecutive safe primes.

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cp's OEIS Frontend

A059323 Smaller term of closest safe prime pairs.

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Magma

Mathematica

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Extensions

A059327 a(n) is smallest safe prime (A005385) such that a(n) + 12*n is the next safe prime, i.e., x = (a(n) - 1)/2 and x + 6*n are closest Sophie Germain primes.

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A059322 First differences of sequence of consecutive safe primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A059327 a(n) is smallest safe prime (A005385) such that a(n) + 12n is the next safe prime, i.e., x = (a(n) - 1)/2 and x + 6n are closest Sophie Germain primes.