A023208 -id:A023208 - cp's OEIS frontend

A136076 Father primes of order 7.

59, 89, 179, 269, 359, 449, 479, 569, 659, 719, 809, 929, 1019, 1109, 1259, 1559, 1619, 1709, 1979, 2069, 2099, 2459, 2609, 2699, 2729, 2879, 2909, 2969, 2999, 3359, 3449, 3779, 4049, 4079, 4229, 4259, 4409, 4679, 5309, 5399, 5519, 5849, 6029, 6299, 6329

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075.

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

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136077, A136078, A136079, A136080.

n = 7; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136077 Father primes of order 8.

Original entry on oeis.org

67, 101, 509, 1019, 1223, 1427, 1733, 2243, 2549, 2957, 4079, 4283, 4793, 5303, 6833, 7547, 7649, 7853, 8363, 8669, 9587, 9689, 11117, 11933, 12239, 12647, 12953, 15809, 16217, 18869, 19583, 20297, 20807, 21419, 21929, 22133, 23357, 24683, 25703

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075. For father primes of order 7 see A136075.

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

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136078, A136079, A136080.

n = 8; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136078 Father primes of order 9.

Original entry on oeis.org

113, 151, 227, 379, 569, 607, 797, 911, 1291, 1367, 1709, 1861, 2089, 2621, 2659, 2887, 3001, 3191, 3457, 3761, 4027, 4597, 4787, 5167, 5281, 5851, 5927, 6421, 6991, 7219, 7561, 7789, 8017, 9689, 10601, 10867, 11171, 11399, 11437, 11551, 11779

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075. For father primes of order 7 see A136076. For father primes of order 8 see A136077.

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

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136079, A136080.

n = 9; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a
Select[Prime[Range[1500]],PrimeQ[(#-18)/19]&] (* Harvey P. Dale, Mar 26 2023 *)

A136079 Father primes of order 10.

Original entry on oeis.org

83, 167, 251, 293, 419, 503, 797, 881, 1259, 1301, 1427, 1511, 1553, 1889, 2141, 2267, 2309, 2393, 2687, 2897, 2939, 3191, 3527, 3779, 3821, 4073, 4157, 4451, 4703, 4787, 5039, 5081, 5417, 5669, 5711, 6173, 6551, 6971, 7307, 7349, 7433, 7559, 7727, 7853

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n, see A136026 (also definition). For father primes of orders 1,2,...,9, see A094524, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, respectively.
From Bob Selcoe, Apr 25 2014: (Start)
In general, a father prime, p', of order k is of the form p'=2k+(2k+1)*p for some prime, p. In this sequence, k=10, and so each prime is of the form p'=20+21p where p ranges over {3,7,11,13,19,23,...}. Thus a father prime p' has order k when (p'-2k)/(2k+1) is prime.
Father primes (p') of order k will be of the form: p'(mod (4k+2))=4k+1, or p'=(4k+2)*j-1, j>=2. For this sequence: k=10, 4k+2=42; j={2,4,6,7,10,12,...}. So for example, j=7 generates a father prime because 42*7-1 = 293 AND (293-(2*10))/(2*10+1) = 13, since both 13 and 293 are prime. Note that not all j such that (4k+2)*j-1 is prime will produce a father prime. In this example, when j=11, 42*11-1=461 (prime); but (461-(2*10))/(2*10+1) = 21 (not prime). (End)

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

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, A136080.

n = 10; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136080 Father primes of order 11.

Original entry on oeis.org

137, 1103, 1931, 2069, 2621, 3449, 3863, 4001, 4139, 5381, 5519, 6761, 8831, 8969, 10211, 13109, 13523, 13799, 15731, 18353, 19319, 21803, 23321, 23459, 25253, 25391, 28151, 28289, 28979, 29531, 29669, 31601, 32429, 32843, 33119, 34361

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075. For father primes of order 7 see A136076. For father primes of order 8 see A136077. For father primes of order 9 see A136078. For father primes of order 10 see A136079.

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

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, A136079.

n = 11; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A115058 Primes p such that 3*p+2 is not prime.

Original entry on oeis.org

2, 11, 31, 41, 47, 53, 61, 67, 71, 73, 101, 107, 109, 113, 131, 137, 151, 157, 179, 181, 191, 193, 211, 223, 229, 241, 251, 263, 271, 277, 281, 283, 307, 311, 331, 347, 359, 373, 379, 389, 401, 421, 431, 443, 449, 461, 463, 467, 487, 491, 509, 521, 541, 547

Offset: 1

Cino Hilliard, Feb 28 2006

Complement of A023208 in the prime numbers A000040.

Primes p that are also the largest prime factor of p*(p^2-1)*(3*p+2)/24.

			p=11, p*(p^2-1)*(3*p+2)/24 = 1925 = 5*5*7*11.

Number Theory, George E. Andrews 1971, Dover Publications New York, p 4.

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

[p: p in PrimesUpTo(600) | not IsPrime(3*p + 2)]; // Vincenzo Librandi, Apr 17 2013

a={};Do[p=Prime[n];If[ !PrimeQ[3*p+2],AppendTo[a,p]],{n,101}];a (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[100]], ! PrimeQ[3 # + 2]&] (* Vincenzo Librandi, Apr 17 2013 *)

Edited by Max Alekseyev, Feb 06 2010

A229966 Numbers n such that A229964(n) = 3.

Original entry on oeis.org

12, 14, 22, 27, 33, 57, 85, 161, 203, 533, 689, 901, 1121, 1633, 2581, 4181, 5513, 5633, 7439, 10561, 18023, 18881, 20833, 21389, 23941, 25043, 28421, 32033, 37733, 48641, 58241, 64643, 66901, 77423, 80033, 84001, 90133, 106439, 116821, 119201, 149189, 155041

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {12, 14, 22, 27, 57} UNION {pq | p, q prime, q = 3p+2 or (p >= 5 and q = 4p+1)}.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229965, A229967, A023208, A023212.

[p * (3*p+2) for p in prime_range(10000) if (3*p+2).is_prime()] + [p * (4*p+1) for p in prime_range(5, 10000) if (4*p+1).is_prime()] + [12, 14, 22, 27, 57]

A023246 Primes that remain prime through 2 iterations of the function f(x) = 3*x + 2.

Original entry on oeis.org

5, 7, 19, 29, 79, 89, 97, 127, 139, 167, 317, 337, 397, 419, 607, 659, 709, 877, 929, 1069, 1129, 1409, 1699, 1777, 2029, 2099, 2267, 2339, 2557, 2617, 2707, 2837, 2917, 2939, 3019, 3067, 3389, 3407, 3529, 3617, 3659, 3719, 4229, 4549, 4919, 5209, 5227, 5417

Offset: 1

David W. Wilson

nonn

Primes p such that 3*p+2 and 9*p+8 are also primes. - Vincenzo Librandi, Aug 04 2010

All terms after the first == 7 or 9 (mod 10). - Robert Israel, Sep 12 2016

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

Subsequence of A023208.

[n: n in [0..100000] | IsPrime(n) and IsPrime(3*n+2) and IsPrime(9*n+8)] // Vincenzo Librandi, Aug 04 2010

select(t -> isprime(t) and isprime(3*t+2) and isprime(9*t+8), [5, seq(seq(10*i+j,j=[7,9]),i=0..10^4)]); # Robert Israel, Sep 12 2016

Select[Prime[Range[750]],And@@PrimeQ[Rest[NestList[3#+2&,#,2]]]&] (* Harvey P. Dale, May 05 2014 *)

A106067 Primes p such that 3p + 2 and 2p + 3 are primes.

Original entry on oeis.org

5, 7, 13, 17, 19, 29, 43, 89, 97, 127, 139, 167, 173, 197, 199, 227, 269, 337, 349, 353, 383, 397, 409, 439, 503, 523, 607, 643, 659, 797, 859, 887, 929, 1013, 1039, 1063, 1069, 1109, 1153, 1193, 1259, 1277, 1303, 1307, 1427, 1429, 1483, 1559, 1567, 1583

Offset: 1

Zak Seidov, May 07 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Intersection of A023204 and A023208.

[p: p in PrimesUpTo(10000)| IsPrime(3*p+2) and IsPrime(2*p+3)]; // Vincenzo Librandi, Nov 13 2010

Select[Prime[Range[20000]], PrimeQ[2#+3]&&PrimeQ[3#+2]&]

is(p) = isprime(p) && isprime(3*p+2) && isprime(2*p+3); \\ Amiram Eldar, Nov 08 2024

A265763 Numerators of primes-only best approximates (POBAs) to 3; see Comments.

Original entry on oeis.org

7, 5, 17, 13, 23, 19, 31, 41, 37, 53, 59, 71, 67, 89, 113, 109, 131, 127, 139, 157, 179, 181, 199, 211, 239, 251, 269, 293, 311, 307, 337, 383, 379, 409, 419, 449, 491, 487, 503, 499, 521, 541, 571, 577, 593, 599, 631, 683, 701, 719, 751, 773, 769, 787, 809

Offset: 1

Clark Kimberling, Dec 18 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

			The POBAs for 3 start with 7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.

Cf. A000040, A023208, A088878, A091180, A094525, A222565.

x = 3; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265763/A265764 *)
Numerator[tL]   (* A091180 *)
Denominator[tL] (* A088878 *)
Numerator[tU]   (* A094525 *)
Denominator[tU] (* A023208 *)
Numerator[y]    (* A265763 *)
Denominator[y]  (* A265764 *)

cp's OEIS Frontend

A136076 Father primes of order 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A136077 Father primes of order 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A136078 Father primes of order 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A136079 Father primes of order 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A136080 Father primes of order 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A115058 Primes p such that 3*p+2 is not prime.

Views

Author

Keywords

Comments

Examples

References

Links

Programs

Magma

Mathematica

Extensions

A229966 Numbers n such that A229964(n) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Sage

A023246 Primes that remain prime through 2 iterations of the function f(x) = 3*x + 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A106067 Primes p such that 3*p + 2 and 2*p + 3 are primes.

Views

A106067 Primes p such that 3p + 2 and 2p + 3 are primes.