A005384 -id:A005384 - cp's OEIS frontend

A065406 Mersenne prime exponents (A000043) which are also Sophie Germain primes (A005384).

2, 3, 5, 89, 9689, 21701, 859433, 43112609

Offset: 1

Labos Elemer, Nov 06 2001

From Gord Palameta, Jul 19 2018: (Start)
All terms after the first two are congruent to 1 modulo 4, because if p is a Sophie Germain prime that is congruent to 3 modulo 4 then 2p + 1 divides 2^p - 1.
Boklan and Conway conjecture that this sequence is finite.
(End)

			31 = 2^5 - 1 and 11 = 2 * 5 + 1 are primes.

David W. Ash, Ian F. Blake, and Scott A. Vanstone, Low Complexity Normal Bases, Discrete Applied Mathematics, 25(1989), 206.
Kent D. Boklan, John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016, p. 6.

Cf. A000043, A005384.

Select[Prime[Range[1000]], PrimeQ[2# + 1] && PrimeQ[2^# - 1] &] (* Alonso del Arte, Jul 20 2018 *)
Select[Prime@ Range[10^6], And[PrimeQ[2 # + 1], MersennePrimeExponentQ@ #] &] (* Michael De Vlieger, Jul 20 2018 *)

a(8) = 43112609, since the ordinal position of this term in A000043 is now confirmed. - Gord Palameta, Jul 19 2018

A066818 a(n) is the least k such that n + Sum_{i=1..k} A005384(i) is prime; or 0 if none exists.

Original entry on oeis.org

1, 2, 1, 12, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 4, 5, 2, 7, 2, 1, 6, 1, 6, 3, 2, 3, 6, 1, 2, 3, 2, 1, 4, 1, 2, 3, 8, 1, 4, 11, 2, 3, 4, 1, 4, 5, 2, 13, 2, 1, 4, 1, 8, 3, 2, 3, 6, 1, 2, 7, 2, 1, 10, 1, 8, 3, 2, 15, 4, 1, 2, 3, 4, 1, 4, 5, 2, 7, 4

Offset: 1

Joseph L. Pe, Jan 19 2002

nonn

There is some empirical evidence to suggest a(n) is nonzero for every n. That is, every n can be expressed as the difference between a prime and a partial sum of the Sophie Germain primes series. See A066753 for a similar conjecture.

			7 + (2 + 3 + 5) = 17, a prime and three consecutive Sophie Germain primes starting from 2, the first Sophie Germain prime, are needed to achieve this. So a(7) = 3.

Cf. A005384, A066753.

a(n) = my(p=0, s=n); for(k=1, oo, until(isprime(2*p+1), p=nextprime(p+1)); if(isprime(s+=p), return(k))); \\ Jinyuan Wang, Jul 30 2020

a(53) corrected by and more terms from Jinyuan Wang, Jul 30 2020

A080209 Gilbreath transform of the sequence of Sophie Germain primes (A005384), i.e., the diagonal of leading successive absolute differences of A005384.

Original entry on oeis.org

2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3

Offset: 1

John W. Layman, Mar 20 2003

nonn

Conjecture: The diagonal of leading successive absolute differences of the Sophie Germain primes consists, except for the initial 2, only of 1's and 3s.

			The difference table begins:
   2;
   3,  1;
   5,  2,  1;
  11,  6,  4,  3;
  23, 12,  6,  2,  1;
  29,  6,  6,  0,  2,  1;

Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Sophie Germain Prime.
Eric Weisstein's World of Mathematics, Gilbreath's Conjecture

Cf. A005384, A036262, A054977.

sgp[1] = Select[Prime[Range[1000]], PrimeQ[2 # + 1]&];
sgp[n_] := Differences[sgp[n - 1]] // Abs;
Table[sgp[n], {n, 1, 105}][[All, 1]] (* Jean-François Alcover, Feb 04 2019 *)

A095750 "Degree" of the Sophie Germain primes (A005384).

Original entry on oeis.org

0, 0, 1, 2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 0

Andrew S. Plewe, Jul 09 2004

This sequence is derived from the special case of Cunningham chains of the first kind where every member of the chain is a Sophie Germain prime.

This sequence can be obtained by subtracting 2 from A074313 and then deleting all negative members. - David Wasserman, Sep 13 2007

			Entries 0, 0, 1, 2, 3 correspond to the Sophie Germain primes 2, 3, 5, 11, 23. 5 is degree 1 because 5 = (2 * 2) + 1 and 2 is also a Sophie Germain prime. Similarly, 11 = (5 * 2) + 1, therefore 11 is degree 2. 23 = (11 * 2) + 1, thus 23 is degree 3 and so on.

C. K. Caldwell, Cunningham Chains.
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A005384.

More terms from David Wasserman, Sep 13 2007

A100031 Bisection of A005384.

Original entry on oeis.org

2, 5, 23, 41, 83, 113, 173, 191, 239, 281, 359, 431, 491, 593, 653, 683, 743, 809, 953, 1019, 1049, 1223, 1289, 1439, 1481, 1511, 1583, 1733, 1889, 1931, 2003, 2063, 2129, 2273, 2351, 2399, 2543, 2693, 2741, 2819, 2939, 2969, 3299, 3359, 3413, 3491, 3593

Offset: 0

N. J. A. Sloane, Nov 20 2004

Cf. A005384.

A:={}: for n from 1 to 608 do if isprime(2*ithprime(n)+1)=true then A:=A union {ithprime(n)} else A:=A fi od: B:=convert(A, list): seq(B[2*j-1],j=1..nops(B)/2);

More terms from Emeric Deutsch, Dec 09 2004

A100032 Bisection of A005384.

Original entry on oeis.org

3, 11, 29, 53, 89, 131, 179, 233, 251, 293, 419, 443, 509, 641, 659, 719, 761, 911, 1013, 1031, 1103, 1229, 1409, 1451, 1499, 1559, 1601, 1811, 1901, 1973, 2039, 2069, 2141, 2339, 2393, 2459, 2549, 2699, 2753, 2903, 2963, 3023, 3329, 3389, 3449, 3539, 3623

Offset: 0

N. J. A. Sloane, Nov 20 2004

A:={}: for n from 1 to 608 do if isprime(2*ithprime(n)+1)=true then A:=A union {ithprime(n)} else A:=A fi od: B:=convert(A, list): seq(B[2*j],j=1..nops(B)/2);

More terms from Emeric Deutsch, Dec 09 2004

A134729 Concatenation of next n Sophie Germain primes A005384(n).

Original entry on oeis.org

2, 35, 112329, 41538389, 113131173179191, 233239251281293359, 419431443491509593641, 653659683719743761809911, 95310131019103110491103122312291289, 1409143914511481149915111559158316011733, 18111889190119311973200320392063206921292141

Offset: 1

Omar E. Pol, Nov 10 2007

Cf. A005384, A053067, A133013, A133014.

With[{sgp=Select[Prime[Range[500]],PrimeQ[2#+1]&]},Table[ FromDigits[ Flatten[IntegerDigits/@Take[sgp,{(n(n+1))/2+1,((n+1)(n+2))/2}]]],{n,0,11}]] (* Harvey P. Dale, Mar 24 2013 *)

More terms from Harvey P. Dale, Mar 24 2013

A134730 Successive digits of Sophie Germain primes A005384(n).

Original entry on oeis.org

2, 3, 5, 1, 1, 2, 3, 2, 9, 4, 1, 5, 3, 8, 3, 8, 9, 1, 1, 3, 1, 3, 1, 1, 7, 3, 1, 7, 9, 1, 9, 1, 2, 3, 3, 2, 3, 9, 2, 5, 1, 2, 8, 1, 2, 9, 3, 3, 5, 9, 4, 1, 9, 4, 3, 1, 4, 4, 3, 4, 9, 1, 5, 0, 9, 5, 9, 3, 6, 4, 1, 6, 5, 3, 6, 5, 9, 6, 8, 3

Offset: 1

Omar E. Pol, Nov 10 2007

Cf. A033308, A005384.

A261963 Smallest number that can be written as the sum of a Sophie Germain prime (A005384) and a practical number (A005153) in exactly n ways.

Original entry on oeis.org

2, 3, 6, 23, 31, 35, 65, 59, 95, 131, 173, 233, 203, 239, 257, 299, 317, 323, 473, 443, 635, 563, 671, 719, 809, 701, 779, 839, 803, 1109, 971, 1049, 1139, 1343, 1103, 1409, 1433, 1607, 1481, 1499, 1559, 1523, 1769, 1679, 1643, 2069, 2063, 2309, 2111, 2141

Offset: 0

Felix Fröhlich, Sep 06 2015

nonn

			23 can be written as the sum of a Sophie Germain prime and a practical number in the following three ways: 3 + 20, 5 + 18, 11 + 12.
Since 23 is the smallest number that can be expressed like that in exactly three ways, a(3) = 23.

Cf. A005153, A005384, A209253.

\\ First define the function is_a005153(n) as in A005153
is_a005384(n) = ispseudoprime(n) && ispseudoprime(2*n+1)
count(n) = x=1; y=n-1; i=0; while(y > n/2, if((is_a005153(x) && is_a005384(y)) || (is_a005153(y) && is_a005384(x)), i++); x++; y--); i
a(n) = k=2; while(count(k)!=n, k++); k

A379148 a(n) is the number of iterations of the function x --> 2*x + 1 such that x remains prime, starting from A005384(n).

Original entry on oeis.org

4, 1, 3, 2, 1, 1, 2, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 3, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Ctibor O. Zizka, Dec 16 2024

nonn

Cunningham chain of the first kind of length i is a sequence of prime numbers (p_1, ..., p_i) such that p_(r + 1) = 2*p_r + 1 for all 1 =< r < i. This sequence tells the length of the Cunningham chain of the first kind for primes from A005384.

			n = 1: A005384(1) = 2 --> 5 --> 11 --> 23 --> 47 --> 95, 95 is not a prime, thus a(1) = 4.
n = 2: A005384(2) = 3 --> 7 --> 15, 15 is not a prime, thus a(2) = 1.

Cf. A000040, A005384, A005385, A371980.

s[n_] := -2 + Length[NestWhileList[2*# + 1 &, n, PrimeQ[#] &]]; Select[Array[s, 5000], # > 0 &] (* Amiram Eldar, Dec 16 2024 *)

a(A371980(n)) = 1.

cp's OEIS Frontend

A065406 Mersenne prime exponents (A000043) which are also Sophie Germain primes (A005384).

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A066818 a(n) is the least k such that n + Sum_{i=1..k} A005384(i) is prime; or 0 if none exists.

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A080209 Gilbreath transform of the sequence of Sophie Germain primes (A005384), i.e., the diagonal of leading successive absolute differences of A005384.

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Mathematica

A095750 "Degree" of the Sophie Germain primes (A005384).

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A100031 Bisection of A005384.

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Maple

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A100032 Bisection of A005384.

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Maple

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A134729 Concatenation of next n Sophie Germain primes A005384(n).

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Mathematica

Extensions

A134730 Successive digits of Sophie Germain primes A005384(n).

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A261963 Smallest number that can be written as the sum of a Sophie Germain prime (A005384) and a practical number (A005153) in exactly n ways.

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Author

Keywords

Examples

Crossrefs

Programs

PARI

A379148 a(n) is the number of iterations of the function x --> 2*x + 1 such that x remains prime, starting from A005384(n).

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