A005384 -id:A005384

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A059453

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

+20
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

Introducing terms of Cunningham chains of first kind.

LINKS

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

Chris K. Caldwell, Cunningham Chains.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(Python)

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

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

CROSSREFS

Intersection of A005384 and A059456.

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

KEYWORD

nonn

AUTHOR

Labos Elemer, Feb 02 2001

STATUS

approved

A156541

Multiplicative closure of Sophie Germain primes (A005384).

+20
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

Subsequence of A130176.

LINKS

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

MATHEMATICA

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

CROSSREFS

Cf. A156541, A156543.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Feb 10 2009

STATUS

approved

A172037

Prime partial sums of Sophie Germain primes A005384.

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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}.}.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jan 23 2010

EXTENSIONS

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

STATUS

approved

A074259

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

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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.)

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

First differences of A005384.

KEYWORD

nonn

AUTHOR

Jon Perry, Sep 20 2002

EXTENSIONS

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

STATUS

approved

A099108

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..55.

CROSSREFS

Cf. A005382, A005384, A099109.

KEYWORD

nonn

AUTHOR

Pierre CAMI, Sep 27 2004

STATUS

approved

A099109

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

3 and 5 are excluded as being trivial solutions.

LINKS

Table of n, a(n) for n=1..39.

CROSSREFS

For n values see A099108

Cf. A005382 A005384 A099108.

KEYWORD

nonn

AUTHOR

Pierre CAMI, Sep 27 2004

STATUS

approved

A173897

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

J. S. Cheema, Table of n, a(n) for n = 1..10000

Wikipedia, Sophie Germain Primes

EXAMPLE

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

PROG

(Sage) 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

(PARI) 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

CROSSREFS

Cf. A005384.

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

KEYWORD

nonn

AUTHOR

Jaspal Singh Cheema, Mar 01 2010

EXTENSIONS

Edited by D. S. McNeil, Dec 02 2010

STATUS

approved

A276821

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

+20
1

2, 2, 29, 3299, 4866623, 22081407211439

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..6.

EXAMPLE

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).

CROSSREFS

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

KEYWORD

nonn

AUTHOR

M. F. Hasler, Sep 18 2016

STATUS

approved

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).

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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?"

LINKS

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.

FORMULA

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)).

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(Python)

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

CROSSREFS

Cf. A005384.

KEYWORD

nonn

AUTHOR

A.H.M. Smeets, Mar 07 2020

STATUS

approved

A360757

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

PROG

(Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [p:p in [1..2500]| IsPrime(Floor(f(p))) and IsPrime(2*Floor(f(p))+1) ];

CROSSREFS

Cf. A003415, A005384.

Subsequence of A157037.

KEYWORD

nonn

AUTHOR

Marius A. Burtea, Mar 01 2023

STATUS

approved

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