A040117 -id:A040117 - cp's OEIS frontend

A243890 Primes of the form 2n^2+38n+17.

101, 149, 257, 317, 449, 521, 677, 761, 941, 1697, 1949, 2081, 2357, 2801, 2957, 3449, 3797, 4349, 4937, 6221, 6449, 6917, 7649, 7901, 8681, 9221, 9497, 10061, 10937, 12161, 13121, 13781, 15149, 16217, 17321, 18077, 18461, 20441, 20849, 25601, 26981, 27449

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A040117.
Conjecture: except 521, 2^a(n)-1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 327 is a square. - Vincenzo Librandi, Jun 29 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A040117.

Cf. similar sequences listed in A243888.

[a: n in [1..200] | IsPrime(a) where a is 2*n^2+38*n+17];

Select[Table[2 n^2 + 38 n + 17, {n, 800}], PrimeQ]

A243891 Primes of the form 2n^2 + 62n + 29.

Original entry on oeis.org

233, 389, 653, 953, 1061, 1289, 1409, 2069, 2213, 4253, 4649, 5273, 6869, 8933, 9209, 10061, 10949, 13829, 15569, 16661, 17033, 17789, 24413, 26693, 28109, 32573, 35729, 36269, 37361, 42473, 44249, 46061, 48533, 51713, 52361, 55661, 56333, 57689, 59753

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A040117.
Conjecture: except 4253, 2^a(n) - 1 is not prime; in other words, these primes are included in A054723.
2*a(n) + 903 is a square. - Vincenzo Librandi, Jun 29 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A040117.

Cf. similar sequences listed in A243888.

[a: n in [1..200] | IsPrime(a) where a is 2*n^2+62*n+29];

Select[Table[2 n^2 + 62 n + 29, {n, 200}], PrimeQ]

A243958 Primes of the form 2n^2+86n+41.

Original entry on oeis.org

317, 521, 857, 977, 1229, 1361, 1637, 2081, 2237, 2729, 3257, 3821, 4217, 4421, 5501, 6197, 8501, 9341, 9629, 12401, 13397, 14081, 15137, 15497, 16229, 18521, 18917, 20129, 21377, 22229, 23537, 23981, 26261, 26729, 29129, 31121, 32141, 35837, 36929, 39161

Offset: 1

Vincenzo Librandi, Jun 16 2014

Subsequence of A040117.

Conjecture: except 521, 2^a(n)-1 is not prime; in other words, these primes are included in A054723.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A040117.

Cf. similar sequences listed in A243888.

[a: n in [1..300] | IsPrime(a) where a is 2*n^2+86*n+41];

Select[Table[2 n^2 + 86 n + 41, {n, 800}], PrimeQ]

A124986 Primes of the form 12k + 5 generated recursively. Initial prime is 5. General term is a(n) = Min_{p is prime; p divides 1 + 4Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

5, 101, 1020101, 53, 29, 2507707213238852620996901, 449, 433361, 401, 925177698346131180901394980203075088053316845914981, 44876921, 17, 173

Offset: 1

Nick Hobson, Nov 18 2006 and Nov 23 2006

nonn

All prime divisors of 1+4Q^2 are congruent to 1 modulo 4.
At least one prime divisor of 1+4Q^2 is congruent to 2 modulo 3 and hence to 5 modulo 12.
The first seven terms are the same as those of A057207.
The next term is known but is too large to include.

			a(8) = 433361 is the smallest prime divisor congruent to 5 mod 12 of 1+4Q^2 = 3179238942812523869898723304484664524974766291591037769022962819805514576256901 = 13 * 433361 * 42408853 * 2272998442375593325550634821 * 5854291291251561948836681114631909089, where Q = 5 * 101 * 1020101 * 53 * 29 * 2507707213238852620996901 * 449.

Robert Price, Table of n, a(n) for n = 1..14

Cf. A000945, A040117, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={5}; q=1;
For[n=2,n<=5,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4*q^2+1][[All,1]],Mod[#,12]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

A124987 Primes of the form 12k+5 generated recursively. Initial prime is 5. General term is a(n) = Min {p is prime; p divides 4+Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

5, 29, 17, 6076229, 1289, 78067083126343039013, 521, 8606045503613, 15837917, 1873731749, 809, 137, 2237, 17729

Offset: 1

Nick Hobson, Nov 18 2006

Since Q is odd, all prime divisors of 4+Q^2 are congruent to 1 modulo 4.
At least one prime divisor of 4+Q^2 is congruent to 2 modulo 3 and hence to 5 modulo 12.
The first two terms are the same as those of A057208.

			a(3) = 17 is the smallest prime divisor congruent to 5 mod 12 of 4+Q^2 = 21029 = 17 * 1237, where Q = 5 * 29.

Tyler Busby, Table of n, a(n) for n = 1..15

Cf. A000945, A040117, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a={5}; q=1;
For[n=2,n<=5,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[q^2+4][[All,1]],Mod[#,12]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

A180217 a(n) = (n-th prime modulo 3) + (n-th prime modulo 4).

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Jan 16 2011

nonn

a(n) = 2 iff prime(n) == 1 (mod 12); a(n) = 2 for prime(n) = 13, 37, 61, 73, 97, 109, ... (A068228).
a(n) = 5 iff prime(n) == 11 (mod 12); a(n) = 5 for prime(n) = 11, 23, 47, 59, 71, 83, ... (A068231).
For n > 2, a(n) = 3 iff prime(n) == 5 (mod 12); a(n) = 3 for prime(n) = 5, 17, 29, 41, 53, 89, ... (A040117).
For n > 2, a(n) = 4 iff prime(n) == 7 (mod 12); a(n) = 4 for prime(n) = 7, 19, 31, 43, 67, 79, ... (A068229).

Equals A039701 + A039702.

Cf. A068228, A068231, A040117, A068229, A039710.

A180217:=func< n | p mod 3 + p mod 4 where p is NthPrime(n) >; [ A180217(n): n in [1..105] ]; // Klaus Brockhaus, Jan 18 2011

Mod[#,3]+Mod[#,4]&/@Prime[Range[110]] (* Harvey P. Dale, Nov 09 2011 *)

A274465 Primes which are the sum of cousin prime pairs - 1.

Original entry on oeis.org

17, 29, 41, 89, 137, 197, 257, 389, 449, 461, 557, 617, 701, 761, 797, 881, 929, 977, 1229, 1289, 1481, 1709, 1721, 1877, 2609, 2861, 2897, 2969, 3137, 3221, 3329, 3389, 3989, 4001, 4409, 4481, 4877, 5081, 5237, 5381, 5417, 5501, 5669, 5717, 6329, 6689, 6917, 7229

Offset: 1

Debapriyay Mukhopadhyay, Jun 24 2016

nonn

Cousin primes are prime pairs that differ by 4. Any prime p in this sequence is such that p = (p-3)/2 + (p+5)/2 - 1, where (p-3)/2 and (p+5)/2 are also primes and they differ by 4.
Proper subset of A040117 (e.g., 5 isn't in the sequence). - David A. Corneth, Jun 24 2016
Intersection of A145471 and A089531. - Michel Marcus, Jun 27 2016
Subsequence of A072669. - Michel Marcus, Jun 27 2016

			17 = 7 + 11 - 1. Note that, (17-3)/2 = 7 and (17+5)/2 = 11 and 7, 11 are cousin prime pairs.
29 = 13 + 17 - 1. Note that, (29-3)/2 = 13 and (29+5)/2 = 17 and 13, 17 are cousin prime pairs.
41 = 19 + 23 - 1. Note that, (41-3)/2 = 19 and (41+5)/2 = 23 and 19, 23 are cousin prime pairs.
89 = 43 + 47 - 1. Note that, (89-3)/2 = 43 and (89+5)/2 = 47 and 43, 47 are cousin prime pairs.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A023200, A152091, A040117, A145471, A089531, A072669.

Select[2 # + 3 &@ Select[Prime@ Range@ 512, PrimeQ[# + 4] &], PrimeQ] (* Michael De Vlieger, Jun 26 2016 *)

is(n) = isprime(n) && isprime((n-3)/2) && isprime((n+5)/2) \\ David A. Corneth, Jun 24 2016

use ntheory ":all"; say for grep{is_prime($)} map { $+$+4-1 } sieve_prime_cluster(1,5e8,4) # _Dana Jacobsen, Apr 27 2017

A347838 Positive numbers that are congruent to 2, 5, or 11 modulo 12.

Original entry on oeis.org

2, 5, 11, 14, 17, 23, 26, 29, 35, 38, 41, 47, 50, 53, 59, 62, 65, 71, 74, 77, 83, 86, 89, 95, 98, 101, 107, 110, 113, 119, 122, 125, 131, 134, 137, 143, 146, 149, 155, 158, 161, 167, 170, 173, 179, 182, 185, 191, 194, 197, 203, 206, 209, 215, 218, 221, 227, 230, 233, 239

Offset: 1

Wolfdieter Lang, Oct 21 2021

This sequence follows from the first column sequence of the array A347834, namely A047529 ({1,3,7} (mod 8)), as given in the formula below.

Together with A017617, the positive integers congruent to 8 modulo 12, one obtains A016789, the positive integers congruent to 2 modulo 3. See the array A347839.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A002264, A016921, A016789, A017617, A017653, A040117(k+1), A047261, A047529, A102283, A347834, A347839.

Map[(3 # + 1)/2 &, LinearRecurrence[{1, 0, 1, -1}, {1, 3, 7, 9}, 60]] (* Michael De Vlieger, Oct 21 2021 *)

a(n) = (3*A047529(n) + 1)/2.
Trisection: a(3*k+1) = 2 + 12*k, a(3*k+2) = 5 + 12*k, a(3*k+3) = 11 + 12*k, or with a(3*k) = -1 + 12*k for k >= 0.
O.g.f. with a(0) =-1: G(x) = (-1 + 3*x + 3*x^2 + 7*x^3)/((1 - x)*(1 - x^3)) = -6/(1-x) + 4/(1-x)^2 + (1 + x)/(1 + x + x^2). Note that (1 - x)*(1 - x^3) = (1-x)^2*(1 + x + x^2) =  1 - x - x^3 + x^4.
a(n) = a(n-1) + a(n-3) - a(n-4), for n >= 4, given a(n) for 0..3, with a(0) = -1.
a(n) = 2*b(n) + 3*b(n-1) + 6*b(n-2) + b(n-3), with b(n) = floor((n+2)/3) = A002264(n+2).
a(n) = -1 + 3*n + 3*floor(n/3) (from the partial fraction decomposition of G).
E.g.f.: 1 + 2*exp(x)*(2*x - 1) + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Dec 08 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = ((sqrt(2)+1)*Pi + sqrt(3)*log(sqrt(3)+2) + sqrt(6)*log(5-2*sqrt(6)))/12. - Amiram Eldar, Dec 30 2021

cp's OEIS Frontend

A243890 Primes of the form 2*n^2+38*n+17.

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A243891 Primes of the form 2*n^2 + 62*n + 29.

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A243958 Primes of the form 2*n^2+86*n+41.

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A124986 Primes of the form 12*k + 5 generated recursively. Initial prime is 5. General term is a(n) = Min_{p is prime; p divides 1 + 4*Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.

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A124987 Primes of the form 12k+5 generated recursively. Initial prime is 5. General term is a(n) = Min {p is prime; p divides 4+Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.

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A180217 a(n) = (n-th prime modulo 3) + (n-th prime modulo 4).

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Magma

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A274465 Primes which are the sum of cousin prime pairs - 1.

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PARI

Perl

A347838 Positive numbers that are congruent to 2, 5, or 11 modulo 12.

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A243890 Primes of the form 2n^2+38n+17.

A243891 Primes of the form 2n^2 + 62n + 29.

A243958 Primes of the form 2n^2+86n+41.

A124986 Primes of the form 12k + 5 generated recursively. Initial prime is 5. General term is a(n) = Min_{p is prime; p divides 1 + 4Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.