A005384 -id:A005384 - cp's OEIS frontend

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

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

A156541 Multiplicative closure of Sophie Germain primes (A005384).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 23, 24, 25, 27, 29, 30, 32, 33, 36, 40, 41, 44, 45, 46, 48, 50, 53, 54, 55, 58, 60, 64, 66, 69, 72, 75, 80, 81, 82, 83, 87, 88, 89, 90, 92, 96, 99, 100, 106, 108, 110, 113, 115, 116, 120, 121, 123, 125, 128, 131, 132

Offset: 1

Reinhard Zumkeller, Feb 10 2009

nonn

A156542(a(n)) = A001221(a(n));

Subsequence of A130176.

R. Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A156541, A156543.

Select[Range@132, And @@ PrimeQ[FactorInteger[#][[All, 1]]*2 + 1] &] (* Ivan Neretin, Aug 30 2015 *)

A172037 Prime partial sums of Sophie Germain primes A005384.

Original entry on oeis.org

2, 5, 73, 167, 2423, 7621, 39233, 50969, 89563, 198139, 207029, 267143, 322963, 335117, 438517, 481207, 541547, 812051, 874697, 917611, 939293, 1077761, 1149593, 1354267, 1464011, 1695559, 1880401, 2510083, 2548703, 3115249, 3157487, 3505849, 4519057

Offset: 1

Jonathan Vos Post, Jan 23 2010

a(1) and a(2) are themselves Sophie Germain primes.

			a(1) = 2 = first Sophie Germain prime A005384(1). a(2) = 5 = sum of first two Sophie Germain primes = 2+3. a(3) = 73 = sum of first six Sophie Germain primes = 2+3+5+11+23+29.

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Cf. A000040, A005384, A005385, A066819, A172036.

Select[Accumulate[Select[Prime[Range[5000]],PrimeQ[2#+1]&]],PrimeQ] (* Harvey P. Dale, Nov 27 2013 *)

A000040 INTERSECTION A066819 = {p such that p is prime and SUM[i=1..k]A005384(k) is prime} = {p such that p is prime and SUM[i=1..k]{p is prime and 2p+1 is prime}.}.

a(7) - a(34) from Nathaniel Johnston, Apr 29 2011

A074259 Gaps between primes p such that 2p+1 is also prime, i.e., Sophie-Germain primes A005384.

Original entry on oeis.org

1, 2, 6, 12, 6, 12, 12, 30, 6, 24, 18, 42, 6, 12, 42, 6, 12, 30, 12, 66, 60, 12, 12, 48, 18, 84, 48, 12, 6, 24, 36, 24, 18, 48, 102, 42, 60, 6, 12, 18, 54, 120, 6, 60, 120, 30, 12, 30, 18, 12, 48

Offset: 1

Jon Perry, Sep 20 2002

nonn

The first two consecutive identical gaps are 12, 12 between A005384(6..8) = (29, 41, 53).

The first three, four and five identical gaps in a row are equal to 30, 150 and 420, respectively, and occur after A005384(85) = 3299, A005384(29952) = 4866623, and A005384(32361449747) = 22081407211439. These were found by N. Fernandez and G. Resta, see link to discussion on the SeqFan mailing list. - M. F. Hasler, Sep 18 2016

M. F. Hasler, Table of n, a(n) for n = 1..7745
Giovanni Resta, in reply to Harvey P. Dale and others, Re: Consecutive Sophie Germain primes with the same gap, SeqFan mailing list, Sep. 2016. (Click "Previous message" to see Neil Fernandez' earlier results.)

First differences of A005384.

Select[Prime[Range[500]],PrimeQ[2#+1]&]//Differences (* Harvey P. Dale, Jul 15 2019 *)

c=0; forprime(p=1+L=2,10^6,if(isprime(2*p+1),write("primegap.txt",c++," "p-L); L=p)) \\ Edited by M. F. Hasler, Sep 16 2016

Edited (name, offset, more terms) by M. F. Hasler, Sep 16 2016

A099108 Numbers n such that A005382(n) + A005384(n) - 1 and A005382(n) + A005384(n) + 1 are twin primes.

Original entry on oeis.org

3, 4, 8, 11, 18, 19, 21, 40, 44, 53, 59, 73, 82, 100, 104, 107, 108, 118, 125, 127, 135, 148, 156, 161, 171, 181, 184, 185, 199, 214, 215, 232, 237, 240, 242, 267, 277, 283, 286, 292, 305, 317, 326, 330, 346, 350, 351, 377, 379, 381, 403, 405, 406, 425, 438

Offset: 1

Pierre CAMI, Sep 27 2004

nonn

1 and 2 are excluded as being trivial solutions ( A005382(1)=A005384(1) and A005382(2)=A005384(2) ).

Cf. A005382, A005384, A099109.

A099109 Primes A005382(n) + A005384(n) - 1 with a twin prime A005382(n) + A005384(n) + 1.

Original entry on oeis.org

11, 29, 149, 311, 617, 659, 857, 2309, 2687, 3671, 4241, 5651, 6569, 8429, 9011, 9281, 9341, 10709, 11549, 11717, 12539, 14321, 15359, 15971, 17291, 18539, 19139, 19211, 21377, 23627, 23909, 26261, 26729, 27479, 27749, 31151, 32801, 33749, 34469

Offset: 1

Pierre CAMI, Sep 27 2004

nonn

3 and 5 are excluded as being trivial solutions.

For n values see A099108

Cf. A005382 A005384 A099108.

A173897 a(n) is the number of Sophie Germain primes (A005384) between prime(n)^2 and prime(n+1)^2.

Original entry on oeis.org

1, 2, 2, 4, 1, 7, 2, 5, 9, 2, 8, 9, 2, 10, 12, 12, 4, 16, 7, 6, 14, 11, 19, 16, 10, 6, 11, 9, 11, 49, 11, 18, 6, 43, 10, 21, 18, 15, 25, 21, 7, 43, 11, 19, 12, 53, 55, 18, 9, 20, 35, 9, 50, 31, 32, 28, 4, 38, 23, 15, 65, 74, 17, 12, 27, 90, 38, 63, 13, 29, 38, 51, 46, 39, 27, 38, 47, 28

Offset: 1

Jaspal Singh Cheema, Mar 01 2010

nonn

If you graph a(n) versus n, an interesting pattern emerges. As you go farther along the n-axis, greater are the number of Sophie Germain primes, on average, within each interval obtained. The smallest count of 1 occurs twice: between squares of (2,3) and (11,13). I suspect the number of Sophie Germain primes within each interval will never be zero. If one could prove that there is at least 1 Sophie Germain prime within each interval, this would imply that Sophie Germain primes are infinite.

			For n = 1, we consider the interval [2^2, 3^2], within which is one Sophie Germain prime, 5. Thus a(1) = 1.

J. S. Cheema, Table of n, a(n) for n = 1..10000
Wikipedia, Sophie Germain Primes

Cf. A005384.

Cf. A069482 (prime(n+1)^2 - prime(n)^2). - Zak Seidov, Sep 04 2016

is_a005384(n) = ispseudoprime(2*n+1)
a(n) = my(i=0); forprime(q=prime(n)^2, prime(n+1)^2, if(is_a005384(q) && q < prime(n+1)^2, i++)); i \\ Felix Fröhlich, Sep 04 2016

A173897 = lambda n: len([p for p in prime_range(nth_prime(n)**2, nth_prime(n+1)**2) if is_prime(2*p+1)]) # D. S. McNeil, Dec 02 2010

Edited by D. S. McNeil, Dec 02 2010

A276821 First of n consecutive Sophie Germain primes (A005384: such that 2p+1 is also prime) in arithmetic progression.

Original entry on oeis.org

2, 2, 29, 3299, 4866623, 22081407211439

Offset: 1

M. F. Hasler, Sep 18 2016

nonn

The corresponding safe primes 2p+1 (A005385) are again the first in that sequence to have the same property.

Terms a(5) and a(6) were given, respectively, by Neil Fernandez and Giovanni Resta, on the SeqFan mailing list, cf. links.

			The first two consecutive identical gaps between Sophie Germain primes are 12 and 12 which occur between A005384(6..8) = (29, 41, 53), therefore a(3) = 29.
The first three consecutive identical gaps between Sophie Germain primes are equal to 30 and occur between A005384(85..88) = (3299, 3329, 3359, 3389), therefore a(4) = 3299.
The first four consecutive identical gaps between Sophie Germain primes are equal to 150 and occur between A005384(29952..29956) = (4866623, 4866773, 4866923, 4867073, 4867223), therefore a(5) = 4866623.
The first five consecutive identical gaps between Sophie Germain primes are equal to 420 and occur between A005384(32361449747..32361449752) = (22081407211439, 22081407211859, 22081407212279, 22081407212699, 22081407213119, 22081407213539), therefore a(6) = 22081407211439.
For n=1 and n=2, a(n) is equal to the smallest Sophie Germain prime, A005384(1) = 2, which is the first of two terms (and also one term) "in arithmetic progression" (which means not any restriction for a single term or any two subsequent terms).

Giovanni Resta, in reply to Harvey P. Dale and others, Re: Consecutive Sophie Germain primes with the same gap, SeqFan mailing list, Sep. 2016. (Click "Previous message" to see Neil Fernandez' earlier results.)

Cf. A005384 (Sophie Germain primes), A074259 (gaps between SG primes), A005385 (safe primes: 2p+1 for SG primes p).

A333084 a(n) equals the smallest Sophie Germain prime q such that pi_(p,2p+1)(q,10,(1,3)) - pi_(p,2p+1)(q,10,(3,1)) = n, where pi_(p,2p+1)(q,10,(b,c)) equals the number of Sophie Germain primes A005384(i) such A005384(i) <= q and (A005384(i),A005384(i+1)) == (b,c) (mod 10).

Original entry on oeis.org

11, 41, 191, 281, 431, 2351, 2741, 31721, 32561, 34631, 35291, 36821, 37181, 60761, 62591, 62981, 63671, 64301, 65171, 196541, 238691, 239201, 241781, 244301, 246731, 255191, 310181, 311021, 358331, 358901, 360611, 361481, 363491, 374771, 376241, 427991

Offset: 1

A.H.M. Smeets, Mar 07 2020

nonn

Except for the Sophie Germain primes 2 and 5, all Sophie Germain primes have either 1, 3 or 9 as least significant digit. Excluding 2 and 5, we start at 11. The sequence of the least significant digits of these prime numbers, i.e., A005384, travels to the following graph
Start -> (1)-----(3)
           \     /
            \   /
             \ /
             (9)    .
Pairs (A005384(i) mod 10, A005384(i+1) mod 10) denote the edges, and the trajectory prefers to travel in this graph in clockwise direction as is shown here. Term a(n), for n > 0, is the least Sophie Germain prime where the (n-1)-th net clockwise cycle has been completed and the Sophie Germain prime next to a(n) has 3 as least significant digit. The start is at vertex (1) in the graph, due to the fact that the first Sophie Germain prime after 2, 3 and 5 is 11, i.e., a(1) = 11.
pi_(p,2p+1)(x;10,(1,3)) is the number of outgoing arrows from vertex (1) in clockwise direction in the graph; pi_(p,2p+1)(x;10,(3,1)) is the number of outgoing arrows from vertex (1) in counterclockwise direction in the graph.
For other prime pairs, like prime twins with vertices (1), (7) and (9) for the lesser of a twin pair and clockwise defined by the order (1) -> (7) -> (9), it seems that their trajectories prefer clockwise cycles through similar graphs too, so an open question is, "is the clockwise preference always the case for prime constellation pairs?"

			The sequence starts at 11 so a(1) = 11, because the next Sophie Germain prime after 11 is 23. For 41 the first clockwise cycle is completed, and the next Sophie Germain prime after 41 is 43, so a(2) = 41. For 131 the number of net clockwise cycles is returned to 0, so 131 is not in the sequence. For 191, the number of net clockwise cycles becomes 2, while the next Sophie Germain prime after 191 is 233, so a(3) = 191.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, Proceedings of the National Academy of Sciences of the United States of America, Vol. 113, No. 31 (2016), E4446-E4454.

Cf. A005384.

togo = 35; mx = togo; T = 0 Range[++togo]; T[[1]] = 11; c = 0; q = 17; While[togo > 1, p=q; While[! PrimeQ[2 (q = NextPrime[q]) + 1]]; t = Mod[{p, q}, 10]; If[t == {3, 1}, c--]; If[t == {1, 3}, c++]; If[0 <= c <= mx && T[[c + 1]] == 0, togo--; T[[c + 1]] = p]]; T (* Giovanni Resta, May 07 2020 *)

def IsPrime(n):
    if n < 2:
        return 0
    elif n == 2 or n == 3:
        return 1
    elif n%2 == 0 or n%3 == 0:
        return 0
    else:
        d, dd = 5, 2
        while d*d <= n and n%d != 0:
            d, dd = d+dd, 6-dd
        if d*d <= n:
            return 0
        else:
            return 1
p = 11
ptry = p
cycle = 0
cmax = 0
while cmax < 36:
    ptry = ptry+6
    if IsPrime(ptry) and IsPrime(2*ptry+1):
        pnext = ptry
        if p%10 == 1 and pnext%10 == 3:
            cycle = cycle+1
        if p%10 == 3 and pnext%10 == 1:
            cycle = cycle-1
        if cycle > cmax:
            print(cycle, p)
            cmax = cycle
        p = pnext

n = pi_(p,2p+1)(a(n);10,(1,3)) - pi_(p,2p+1)(a(n);10,(3,1)).
n-1 = pi_(p,2p+1)(a(n);10,(3,9)) - pi_(p,2p+1)(a(n);10,(9,3)).
n-1 = pi_(p,2p+1)(a(n);10,(9,1)) - pi_(p,2p+1)(a(n);10,(1,9)).

A360757 Numbers k for which the arithmetic derivative of k is a Sophie Germain prime (A005384).

Original entry on oeis.org

6, 42, 154, 182, 222, 231, 286, 357, 434, 442, 455, 483, 582, 595, 645, 690, 742, 762, 770, 806, 861, 906, 969, 987, 994, 1045, 1066, 1086, 1122, 1162, 1463, 1534, 1547, 1554, 1582, 1738, 1742, 1771, 1798, 1869, 1905, 2065, 2121, 2193, 2265, 2274, 2282, 2365

Offset: 1

Marius A. Burtea, Mar 01 2023

nonn

			6' = 5 is prime and 2*6' + 1 = 2*5 + 1 = 11 is prime, so 6 is a term.
42' = 41 is prime and 2*42' + 1 = 2*41 + 1 = 83 is prime, so 42 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003415, A005384.

Subsequence of A157037.

f:=func; [p:p in [1..2500]| IsPrime(Floor(f(p))) and IsPrime(2*Floor(f(p))+1) ];

filter:= proc(n) local np,t;
  np:= n*add(t[2]/t[1], t = ifactors(n)[2]);
  isprime(np) and isprime(2*np+1)
end proc:
select(filter, [$1..3000]); # Robert Israel, Mar 18 2023

d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[2400], PrimeQ[d1 = d[#]] && PrimeQ[2*d1 + 1] &] (* Amiram Eldar, Mar 01 2023 *)

cp's OEIS Frontend

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

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A156541 Multiplicative closure of Sophie Germain primes (A005384).

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A172037 Prime partial sums of Sophie Germain primes A005384.

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A074259 Gaps between primes p such that 2p+1 is also prime, i.e., Sophie-Germain primes A005384.

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A099108 Numbers n such that A005382(n) + A005384(n) - 1 and A005382(n) + A005384(n) + 1 are twin primes.

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A099109 Primes A005382(n) + A005384(n) - 1 with a twin prime A005382(n) + A005384(n) + 1.

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A173897 a(n) is the number of Sophie Germain primes (A005384) between prime(n)^2 and prime(n+1)^2.

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A276821 First of n consecutive Sophie Germain primes (A005384: such that 2p+1 is also prime) in arithmetic progression.

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A333084 a(n) equals the smallest Sophie Germain prime q such that pi_(p,2p+1)(q,10,(1,3)) - pi_(p,2p+1)(q,10,(3,1)) = n, where pi_(p,2p+1)(q,10,(b,c)) equals the number of Sophie Germain primes A005384(i) such A005384(i) <= q and (A005384(i),A005384(i+1)) == (b,c) (mod 10).

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