A002515 -id:A002515 - cp's OEIS frontend

A214873 Primes p such that 2*p + 1 is also prime and p + 1 is a highly composite number (definition 1).

3, 5, 11, 23, 179, 239, 359, 719, 5039, 55439, 665279, 6486479, 32432399, 698377679, 735134399, 1102701599, 20951330399, 3212537327999, 149602080797769599, 299204161595539199, 2718551763981393634806325317503999

Offset: 1

Arkadiusz Wesolowski, Jul 30 2012

nonn

An equivalent definition of this sequence: odd Sophie Germain primes that differ from a highly composite number by 1.
Intersection of A005384 (Sophie Germain primes) and A072828.
With the exception of 5, a subsequence of A002515 (Lucasian primes).
Except for first two terms, this is a subsequence of A054723.
Except for n = 2, 2*a(n) + 1 is a prime factor of A000225(a(n)) (i.e., 2*23 + 1 divides 2^23 - 1).
Conjecture: for n >= 5, GCD(A000032(a(n)), A000225(a(n))) = 2*a(n) + 1.

			23 is a term because both 23 and 47 are primes and also 24 is a highly composite number.

Amiram Eldar, Table of n, a(n) for n = 1..25
Wikipedia, Sophie Germain prime

Cf. A054723.

lst = {}; a = 0; Do[b = DivisorSigma[0, n + 1]; If[b > a, a = b; If[PrimeQ[n] && PrimeQ[2*n + 1], AppendTo[lst, n]]], {n, 1, 10^6, 2}]; lst

A215850 Primes p such that 2*p + 1 divides Lucas(p).

Original entry on oeis.org

5, 29, 89, 179, 239, 359, 419, 509, 659, 719, 809, 1019, 1049, 1229, 1289, 1409, 1439, 1499, 1559, 1889, 2039, 2069, 2129, 2339, 2399, 2459, 2549, 2699, 2819, 2939, 2969, 3299, 3329, 3359, 3389, 3449, 3539, 3779, 4019, 4349, 4409, 4919, 5039, 5279, 5399, 5639

Offset: 1

Arkadiusz Wesolowski, Aug 24 2012

nonn

An equivalent definition of this sequence: 5 together with primes p such that p == -1 (mod 30) and 2*p + 1 is also prime.
Sequence without the initial 5 is the intersection of A005384 and A132236.
These numbers do not occur in A137715.
From Arkadiusz Wesolowski, Aug 25 2012: (Start)
The sequence contains numbers like 1409 which are in A053027.
a(n) is in A002515 if and only if a(n) is congruent to -1 mod 60. (End)

			29 is in the sequence since it is prime and 59 is a factor of Lucas(29) = 1149851.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
C. K. Caldwell, "Top Twenty" page, Lucas cofactor
Eric Weisstein's World of Mathematics, Lucas Number

Supersequence of A230809. Cf. A000032, A132236.

[5] cat [n: n in [29..5639 by 30] | IsPrime(n) and IsPrime(2*n+1)];

Select[Prime@Range[740], Divisible[LucasL[#], 2*# + 1] &]
Prepend[Select[Range[29, 5639, 30], PrimeQ[#] && PrimeQ[2*# + 1] &], 5]

is_A215850(n)=isprime(n)&!real((Mod(2,2*n+1)+quadgen(5))*quadgen(5)^n) \\ - M. F. Hasler, Aug 25 2012

A231916 Primes p such that 16p^2 + 10p + 1 divides 2^p - 1.

Original entry on oeis.org

11, 179, 239, 431, 5231, 7079, 7211, 13451, 14879, 17939, 26111, 28019, 28499, 30851, 36479, 39779, 40559, 43391, 44699, 45179, 48731, 49919, 50411, 66791, 68111, 74099, 74699, 79151, 79811, 83459, 87299, 96731, 96779, 101399, 102551, 103391, 111959, 116411

Offset: 1

Arkadiusz Wesolowski, Nov 15 2013

nonn

Chris Caldwell, The Prime Glossary, Mersenne number
C. K. Caldwell, "Top Twenty" page, Mersenne cofactor

Subsequence of A002515 and of A231917. Supersequence of A231918.

Cf. A001348, A028993.

Select[Prime@Range[10996], PowerMod[2, #, 16*#^2 + 10*# + 1] == 1 &]

A374914 Primes p == 2, 3 (mod 4) with 2*p+1 prime.

Original entry on oeis.org

2, 3, 11, 23, 83, 131, 179, 191, 239, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2339, 2351, 2399, 2459, 2543, 2699, 2819, 2903, 2939, 2963, 3023, 3299, 3359, 3491

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2024

2 together with Lucasian primes (A002515).

Primes p such that p^(p + 1) == p + 1 (mod 2*p + 1).

			2 is in this sequence because 2^(2 + 1) = 8 and 8 = 3 (mod 2*2 + 1) where 2 prime.

Supersequence of A002515. Subsequence of A374913.

Cf. A374912.

Select[Prime[Range[490]],Mod[#^(#+1),2#+1]==#+1 &] (* Stefano Spezia, Jul 23 2024 *)

list(lim)=my(v=List([2])); forprimestep(p=3,lim\1,4, if(isprime(2*p+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 25 2024

a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

A101787 a(n) = |S(n)| where S(n) = {i : 1 <= i <= n and 4i-1 and 8i-1 are primes}.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

Number of Lucasian primes that do not exceed 4*n-1. [Corrected by Amiram Eldar, May 23 2024]

Cf. A002515, A101788.

Accumulate[Table[Boole[And @@ PrimeQ[{4, 8}*i - 1]], {i, 1, 100}]] (* Amiram Eldar, May 23 2024 *)

Name corrected by Amiram Eldar, May 23 2024

A101788 a(n) = n - A101787(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 52, 53, 54, 54, 55, 56, 57, 58, 59, 60, 61

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

Cf. A002515, A101787.

With[{m = 100}, Range[m] - Accumulate[Table[Boole[And @@ PrimeQ[{4, 8}*i - 1]], {i, 1, m}]]] (* Amiram Eldar, May 23 2024 *)

A188132 Primes p such that p == 3 (mod 4) and 6p+1 is prime.

Original entry on oeis.org

3, 7, 11, 23, 47, 83, 103, 107, 131, 151, 263, 271, 283, 311, 331, 347, 367, 443, 467, 503, 607, 683, 727, 751, 787, 863, 887, 907, 947, 971, 1063, 1091, 1103, 1151, 1171, 1283, 1327, 1423, 1427, 1451, 1487, 1511, 1531, 1567, 1607, 1787, 1811, 1823, 1831, 1847, 1907, 1931, 1987

Offset: 1

M. F. Hasler, Mar 21 2011

nonn

Complement of A188131 in A007693 \ {2}.

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

Cf. A002515, A007693, A188131.

Select[Range[3, 2000, 4], PrimeQ[#] && PrimeQ[6# + 1] &] (* Amiram Eldar, Nov 13 2019 *)

forprime( p=1,1e4, p%4==3 & isprime(p*6+1) & print1(p", "))

A307121 Number of Lucasian primes less than 10^n.

Original entry on oeis.org

1, 4, 19, 100, 581, 3912, 28091, 211700, 1655601, 13286320, 109058381, 911436949, 7731247492

Offset: 1

Rodolfo Ruiz-Huidobro, Mar 26 2019

The Lucasian primes are those Sophie Germain primes of the form 4k + 3. They are interesting because if they are infinite in number, then the sequence of Mersenne numbers with prime exponents contains an infinite number of composite integers.

Conjecture: about half of all Sophie Germain primes are Lucasian primes, and the rest are either 2 or a prime of the form 4k + 1.

			There are 4 Lucasian primes below 10^2: {3,11,23,83}, therefore a(2) = 4.

Simon Davis, Arithmetical sequences for the exponents of composite Mersenne numbers, Notes on Number Theory and Discrete Mathematics 20, no. 1 (2014): 19-26.

Cf. A092816, A002515, A307176.

c = 0; r = 10; s = {}; Do[If[p > r, AppendTo[s, c]; r *= 10]; If[PrimeQ[p] && PrimeQ[2p + 1], c++], {p, 3, 1000003, 4}]; s (* Amiram Eldar, Mar 27 2019 *)
lucSophies = Select[4Range[2500000] - 1, PrimeQ[#] && PrimeQ[2# + 1] &]; Table[Length[Select[lucSophies, # < 10^n &]], {n, 0, 7}]

a(n) = { my(t=0); forprime(p=2,10^n,p%4==3 && ispseudoprime(1+(2*p)) && t++);t } \\ Dana Jacobsen, Mar 28 2019

use ntheory ":all"; sub a { my($n,$t)=(shift,0); forprimes { $t++ if ($&3) == 3 && is_prime(1+($<<1)) } 10**$n; $t; } # Dana Jacobsen, Mar 28 2019

a(9)-a(11) from Amiram Eldar, Mar 27 2019
a(12) from Amiram Eldar, Mar 31 2019
a(13) from Dana Jacobsen, Apr 02 2019

A307176 Number of Sophie Germain primes of the form 4k + 1 less than 10^n.

Original entry on oeis.org

1, 5, 17, 89, 589, 3833, 27940, 211439, 1653257, 13283194, 109058142, 911411528, 7731354496

Offset: 1

Rodolfo Ruiz-Huidobro, Mar 27 2019

Sophie Germain primes can alternatively be Lucasian primes, primes of the form 4k + 1, or, the individual prime 2.

			There are five Sophie Germain Primes of the form 4k + 1 below 10^2: {5, 29, 41, 53, 89}, therefore a(2) = 5.

Cf. A091098, A092816, A002515, A307121, A002144, A005384, A103579.

nonLucSophies = Select[4Range[2500000] + 1, PrimeQ[#] && PrimeQ[2# + 1] &]; Table[Length[Select[nonLucSophies, # < 10^n &]], {n, 0, 7}]

a(n) < A092816(n).
a(n) <= A091098(n) (with equality for n = 1).
a(n) = A092816(n) - A307121(n) - 1.

A350703 a(n) is the least integer k such that (2nk+1) | (2^k-1).

Original entry on oeis.org

3, 18, 5, 9, 15, 50, 40, 16, 7, 156, 60, 25, 180, 102, 113, 81, 10, 50, 29, 159, 51, 56, 24, 36, 47, 90, 337, 72, 55, 106, 33, 102, 780, 28, 117, 25, 155, 540, 60, 104, 223, 1012, 168, 180, 91, 540, 3132, 47, 510, 412, 154, 45, 80, 432, 201, 36, 90, 144, 97, 53, 279, 880

Offset: 1

Karl-Heinz Hofmann, Feb 03 2022

nonn

The formula 2nk+1 is used to find trivial factors of Mersenne(p). Here it is used for all exponents (prime exponents and not prime exponents).
Mersenne primes of A000043 can be found in this sequence too (except for 2). E.g.: a(1, 3, 9, 315, 3855, 13797) = A000043(2..7).
If n mod 4 = 2 then a(n) must be composite.

			a(5) = 15: 2^15 - 1 = 32767; 2*5*15 + 1 = 151; 32767 mod 151 = 0, and there are no numbers < 15 which satisfy the requirement for n = 5.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000043, A002515, A188130, A122095, A188133, A350702.

a[n_] := Module[{k = 1}, While[PowerMod[2, k, 2*n*k + 1] != 1, k++]; k];  Array[a, 62] (* Amiram Eldar, Feb 03 2022 *)

a(n) = my(k=1); while (Mod(2, 2*n*k+1)^k != 1, k++); k; \\ Michel Marcus, Feb 03 2022

def A350703(k,expo):
    while pow(2, expo, 2*k*expo+1) != 1: expo += 1
    return expo
print([A350703(k,1) for k in range(1, 63)])

cp's OEIS Frontend

A214873 Primes p such that 2*p + 1 is also prime and p + 1 is a highly composite number (definition 1).

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A215850 Primes p such that 2*p + 1 divides Lucas(p).

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A231916 Primes p such that 16*p^2 + 10*p + 1 divides 2^p - 1.

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A374914 Primes p == 2, 3 (mod 4) with 2*p+1 prime.

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A101787 a(n) = |S(n)| where S(n) = {i : 1 <= i <= n and 4*i-1 and 8*i-1 are primes}.

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A101788 a(n) = n - A101787(n).

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A188132 Primes p such that p == 3 (mod 4) and 6p+1 is prime.

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A307121 Number of Lucasian primes less than 10^n.

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Perl

A231916 Primes p such that 16p^2 + 10p + 1 divides 2^p - 1.

A101787 a(n) = |S(n)| where S(n) = {i : 1 <= i <= n and 4i-1 and 8i-1 are primes}.

A350703 a(n) is the least integer k such that (2nk+1) | (2^k-1).