A005385 -id:A005385 - cp's OEIS frontend

A059452 Safe primes (A005385) that are not Sophie Germain primes.

7, 47, 59, 107, 167, 227, 263, 347, 383, 467, 479, 503, 563, 587, 839, 863, 887, 983, 1187, 1283, 1307, 1319, 1367, 1487, 1523, 1619, 1823, 1907, 2027, 2099, 2207, 2447, 2579, 2879, 2999, 3119, 3167, 3203, 3467, 3947, 4007, 4079, 4127, 4139, 4259, 4283

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 7, these primes are congruent to 11 modulo 12.

Terminal primes in complete Cunningham chains of first kind, i.e., the chains cannot be continued from these primes.

			347 is a term because 173 is a prime but 695 is not.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chain.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains. [Wayback Machine link]

Intersection of A005385 and A053176.

Cf. A005384, A059452, A059453, A059454, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660.

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)/2],If[ !PrimeQ[2*p+1],AppendTo[lst,p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 24 2009 *)

is(p) = p > 2 && isprime(p) && isprime((p-1)/2) && !isprime(2*p+1); \\ Amiram Eldar, Jul 15 2024

from itertools import count, islice
from sympy import isprime, prime
def A059452_gen(): # generator of terms
    return filter(lambda p:isprime(p>>1) and not isprime(p<<1|1),(prime(i) for i in count(1)))
A059452_list = list(islice(A059452_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156659(a(n))*(1-A156660(a(n))) = 1. - Reinhard Zumkeller, Feb 18 2009

Broken link updated by R. J. Mathar, Apr 12 2010

A059456 Unsafe primes: primes not in A005385.

Original entry on oeis.org

2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Labos Elemer, Feb 02 2001

nonn

A010051(a(n))*(1-A156659(a(n))) = 1; subsequence of A156657. - Reinhard Zumkeller, Feb 18 2009
Also, primes p such that p-1 is a non-semiprime. - Juri-Stepan Gerasimov, Apr 28 2010
Conjecture: From the sequence of prime numbers, let 2 and remove the first data iteration of 2*p+1; leave 3 and remove the prime data by the iteration 2*p+1 and we get the sequence. Example for p=2, remove(5,11,23,47); p=3, remove(7); p=13, p=17, p=19, p=23, remove(47); and so on. - Vincenzo Librandi, Aug 07 2010

			31 is here because (31-1)/2=15 is not prime. 2 and 3 are here because 1/2 and 1 are not prime numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330.

Initial terms for groups in A075712.

Complement[Prime@ Range@ PrimePi@ Max@ #, #] &@ Select[Prime@ Range@ 90, PrimeQ[(# - 1)/2] &] (* Michael De Vlieger, May 01 2016 *)
Select[Prime[Range[100]],PrimeOmega[#-1]!=2&] (* Harvey P. Dale, May 13 2018 *)

is(n)=isprime(n) && !isprime(n\2) \\ Charles R Greathouse IV, May 02 2016

a(n) ~ n log n. - Charles R Greathouse IV, Dec 29 2024

A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

Original entry on oeis.org

2, 3, 29, 41, 53, 89, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 2 and 3 these primes are congruent to 5 or 11 modulo 12.

Introducing terms of Cunningham chains of first kind.

			89 is a term because (89-1)/2 = 44 is not prime, but 2*89 + 1 = 179 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A059456.

Cf. A005385, A053176, A059452, A059453, A059454, A059455, A007700, A005602, A023272, A023302, A023330.

lst={};Do[p=Prime[n];If[ !PrimeQ[(p-1)/2],If[PrimeQ[2*p+1],AppendTo[lst,p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 24 2009 *)
Select[Prime[Range[300]],PrimeQ[2#+1]&&!PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Nov 10 2017 *)

is(p) = isprime(p) && isprime(2*p+1) && if(p > 2, !isprime((p-1)/2), 1); \\ Amiram Eldar, Jul 15 2024

from itertools import count, islice
from sympy import isprime, prime
def A059453_gen(): # generator of terms
    return filter(lambda p:not isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059453_list = list(islice(A059453_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156660(a(n))*(1-A156659(a(n))) = 1. - Reinhard Zumkeller, Feb 18 2009

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

A059394 Special safe primes (from A005385) such that the next prime is also a safe prime.

Original entry on oeis.org

5, 7, 467, 1307, 2447, 5087, 5927, 12527, 18947, 44687, 55079, 78467, 79943, 83207, 93383, 103007, 105143, 111443, 118787, 128879, 143687, 164987, 196907, 204587, 207227, 208787, 229487, 232823, 236507, 257627, 267143, 275987, 289319, 296159

Offset: 1

Labos Elemer, Jan 29 2001

nonn

			For {467,55079,103007,728579,887759,..} safe primes {479,55103,103043,728627,887819} are the next primes, whose differences are 12,24,36,48,60,.. respectively,not necessarily equal to 12, the possible minimum (see A059322).

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

Cf. A005385, A005384, A059322, A059323, A059327.

ss[x_] := PrimeQ[(Prime[x]-1)/2]&&PrimeQ[(Prime[x+1]-1)/2] t=Prime[Flatten[Position[Table[ss[w], {w, 1, 100000}], True]]]

A059395 Smaller of safe prime twins: special safe primes (A005385) p such that the next prime is also the next safe prime and is p+12, i.e., occurs at the closest possible distance, 12.

Original entry on oeis.org

467, 1307, 2447, 5087, 5927, 12527, 18947, 44687, 78467, 83207, 118787, 143687, 164987, 196907, 204587, 207227, 208787, 229487, 236507, 257627, 275987, 297707, 330887, 339827, 367007, 369647, 394007, 454907, 458807, 474347, 534827, 536087

Offset: 1

Labos Elemer, Jan 29 2001

nonn

			The pairs (5,7) and (7,11) are omitted, albeit are both consecutive primes and consecutive safe primes, however their distances (2 and 4) are singular. Cases [467, 439] and [20738027, 20738039] are pairs are both consecutive of consecutive primes and consecutive safe primes in minimal distance=12. The corresponding twins of Sophie Germain primes are [233, 239] or [1369013, 1369019] in distance 6.

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

Cf. A005385, A005384, A058322, A059323, A059327.

safeQ[p_] := PrimeQ[(p-1)/2]; seq={}; c=0; p1 = p2 = 11; q1 = safeQ[p1]; While[c < 30, p2 = NextPrime[p2]; q2 = safeQ[p2]; If[q1 && q2 && p2 == p1 + 12, c++; AppendTo[seq, p1]]; p1 = p2; q1 = q2]; seq (* Amiram Eldar, Jan 13 2020 *)

A075705 Safe primes (A005385) (p and (p-1)/2 are primes) such that 6*p+1 is also prime.

Original entry on oeis.org

5, 7, 11, 23, 47, 83, 107, 263, 347, 467, 503, 863, 887, 1283, 1487, 1823, 1907, 2027, 2063, 2447, 2903, 3203, 3623, 4007, 4127, 4547, 4703, 4787, 5387, 5807, 7523, 7703, 8147, 8423, 11423, 11483, 11807, 12107, 12227, 12647, 12983, 13043, 13163, 14087, 14207

Offset: 1

Jani Melik, Oct 02 2002

nonn

			47 is prime, so is (47-1)/2=23 and also 6*47+1=283; 83 is a prime, (83-1)/2=41 and 6*83+1=499, ...

David Consiglio, Jr. and T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A059455, A007693.

ts_sg_var_pras := proc(nmax) local i,tren,atek; tren := 0: for i from 1 to nmax do atek := numtheory[safeprime](i): if (atek > tren) then if (isprime(atek)='true' and isprime(6*atek+1)='true') then tren := atek: fi; fi; od; end: seq(ts_sg_var_pras(i), i=1..3000);

Select[Range[20000], PrimeQ[#] && PrimeQ[(#-1)/2] && PrimeQ[6#+1] &] (* T. D. Noe, Nov 07 2011 *)
Select[Prime[Range[1700]],And@@PrimeQ[{(#-1)/2,6#+1}]&] (* Harvey P. Dale, Feb 28 2013 *)

A075706 Safe primes (A005385) (p and (p-1)/2 are primes) such that 8*p+1 (A023228) is also prime.

Original entry on oeis.org

5, 11, 107, 179, 347, 479, 1187, 1307, 1367, 1487, 1619, 2027, 2207, 2447, 2999, 3119, 3467, 4007, 4079, 4139, 4799, 5087, 5807, 5927, 5939, 6827, 7079, 7247, 8699, 9587, 9839, 10607, 12107, 12539, 12659, 14207, 15299, 16139, 16187, 17027

Offset: 1

Jani Melik, Oct 02 2002

nonn

			11 is a prime, so is (11-1)/2=5 and also 8*11+1=89; 107 is a prime, (107-1)/2=53 and 8*107+1=857, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A059455, A023228.

ts_sg8_var_pras := proc(nmax) local i,tren,atek; tren := 0: for i from 1 to nmax do atek := numtheory[safeprime](i): if (atek > tren) then if (isprime(atek)='true' and isprime(6*atek+1)='true') then tren := atek: fi; fi; od; end: seq(ts_sg8_var_pras(i), i=1..3000);

Select[Prime[Range[2000]],AllTrue[{(#-1)/2,8#+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 31 2020 *)

forprime(p=3,20000,if(isprime((p-1)/2),if(isprime(8*p+1),print1(p","))))

More terms from Ralf Stephan, Mar 19 2003

A152952 Von Staudt primes which are not safe primes (A005385).

Original entry on oeis.org

239, 443, 647, 659, 827, 1223, 1259, 1499, 1787, 1847, 2087, 2243, 2339, 2687, 2699, 3299, 3659, 3767, 4943, 5903, 6263, 6287, 6299, 6563, 6863, 6959, 7043, 7487, 7583, 7883, 7907, 7919, 8087, 8219, 8243, 8387, 8627, 8663

Offset: 1

Peter Luschny, Dec 25 2008

nonn

			239 is a von Staudt prime because the denominator(B(239-1)/(239-1))=239*12, where B(n) is the Bernoulli number, but (239-1)/2=119=7*17 is not a prime.

Jean-François Alcover, Table of n, a(n) for n = 1..100
P. Luschny, Von Staudt prime number, definition and computation.

Cf. A092307, A005385 and A152951.

a := proc(n) local k,L; L:= []; for k from 11 by 12 to n do map(i->i+1,divisors(k-1)); select(isprime,%) minus {2,3}; if % = {k} then L := [op(L),k] fi; od; select(isprime,map(i->i+i+1,select(isprime,[$1..iquo(n,2)]))): sort(convert(convert(L,set) minus convert(%,set),list)): end:

vonStaudtPrimeQ[p_?PrimeQ] := Denominator[BernoulliB[p-1]/(p-1)] == 12*p; safePrimeQ[p_?PrimeQ] := PrimeQ[(p-1)/2]; Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[vonStaudtPrimeQ[p] && !safePrimeQ[p], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Jan 27 2014 *)

A172036 Prime partial sums of safe primes A005385.

Original entry on oeis.org

5, 23, 509, 2267, 9157, 26437, 44357, 55921, 82301, 184843, 276173, 611069, 732271, 757577, 1006559, 1067611, 1195547, 2132113, 2576683, 3243511, 3302393, 4258091, 5530033, 7326931, 7984121, 10518353, 10748449, 10864151, 11096587, 11937257

Offset: 1

Jonathan Vos Post, Jan 23 2010

a(2) = 23 is not only the 3rd partial sum of safe primes, but is itself the 4th safe prime. What are the next safe prime partial sums of safe primes [no more through 66869(41)]?

			a(1) = 5 = A066869(1) is prime. a(2) = 23 = A066869(3) is prime. a(3) = 2267 = A066869(15) is prime.

Cf. A005384, A005385, A066869.

A000040 INTERSECTION A066869 = A000040 INTERSECTION = SUM[i=1..n]A005385 (i) = {p such that p prime and (p-1)/2 prime and p is an element of SUM[i=1..n]{p prime and (p-1)/2 prime}.}.

Corrected and extended by Max Alekseyev, Apr 22 2010

cp's OEIS Frontend

A059452 Safe primes (A005385) that are not Sophie Germain primes.

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A059456 Unsafe primes: primes not in A005385.

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A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

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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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A059394 Special safe primes (from A005385) such that the next prime is also a safe prime.

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A059395 Smaller of safe prime twins: special safe primes (A005385) p such that the next prime is also the next safe prime and is p+12, i.e., occurs at the closest possible distance, 12.

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Mathematica

A075705 Safe primes (A005385) (p and (p-1)/2 are primes) such that 6*p+1 is also prime.

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A075706 Safe primes (A005385) (p and (p-1)/2 are primes) such that 8*p+1 (A023228) is also prime.

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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.