A280720 -id:A280720 - cp's OEIS frontend

A023217 Primes p such that 5*p + 2 is also prime.

3, 7, 13, 19, 31, 61, 67, 73, 79, 97, 109, 151, 157, 181, 193, 199, 223, 313, 331, 349, 373, 397, 457, 487, 523, 541, 571, 577, 607, 613, 643, 661, 691, 709, 727, 739, 769, 811, 859, 919, 991, 997, 1021, 1033, 1039, 1069, 1087, 1129, 1171, 1201, 1213, 1249, 1279, 1321

Offset: 1

David W. Wilson

Except for the first term, all terms are congruent to 1 (mod 6). - John Cerkan, Sep 07 2016
Numbers k such that A280720(k) > 0. - Felix Fröhlich, Jan 07 2017
Intersection of A000040 and A111223. - Felix Fröhlich, Jan 07 2017

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

Cf. A000040, A111223, A280720. Subsequence of A111223.

[n: n in PrimesUpTo(400) | IsPrime(5*n+2)]; // Vincenzo Librandi, Nov 20 2010

A023217:=n->`if`(isprime(n) and isprime(5*n+2), n, NULL): seq(A023217(n), n=1..3*10^3); # Wesley Ivan Hurt, Sep 07 2016

Select[Prime[Range[200]], PrimeQ[5 # + 2] &] (* Vincenzo Librandi, May 20 2014 *)

is(n) = ispseudoprime(n) && ispseudoprime(5*n+2) \\ Felix Fröhlich, Jan 07 2017

A023313 Primes that remain prime through 4 iterations of function f(x) = 5x + 2.

Original entry on oeis.org

373, 1171, 13687, 21997, 25657, 61603, 74413, 76471, 84199, 87181, 93487, 114691, 135859, 170761, 174877, 184333, 192979, 196177, 207931, 209743, 244219, 276229, 286687, 292561, 297811, 334603, 338893, 405037, 408361, 417097, 439141, 446323

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+2, 25*p+12, 125*p+62 and 625*p+312 are also primes. - Vincenzo Librandi, Aug 04 2010

Numbers k such that A280720(k) > 3. - Felix Fröhlich, Jan 07 2017

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

Cf. A280720. Subsequence of A023217, A023252, A023283, and A111223.

[n: n in [1..1000000] | IsPrime(n) and IsPrime(5*n+2) and IsPrime(25*n+12) and IsPrime(125*n+62) and IsPrime(625*n+312)]; // Vincenzo Librandi, Aug 04 2010

Select[Range[10^6], Times @@ Boole@ Map[PrimeQ, NestList[5 # + 2 &, #, 4]] == 1 &] (* Michael De Vlieger, Jan 09 2017 *)

a(n) == 31 or 37 (mod 42). - John Cerkan, Oct 07 2016

A023341 Primes that remain prime through 5 iterations of function f(x) = 5x + 2.

Original entry on oeis.org

135859, 174877, 192979, 244219, 292561, 679297, 842341, 964897, 1076029, 1470241, 1990579, 2004943, 2339263, 2615707, 2625577, 2633557, 2892277, 3003787, 3201901, 3758233, 4406797, 5065861, 5157547, 5390857, 5424961, 5546173, 5875369, 7746217

Offset: 1

David W. Wilson

nonn

Primes p such that 5*p+2, 25*p+12, 125*p+62, 625*p+312 and 3125*p+1562 are also primes. - Vincenzo Librandi, Aug 05 2010

Numbers k such that A280720(k) > 4. - Felix Fröhlich, Jan 07 2017

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

Subsequence of A023217, A023252, A023283, A023313, and A111223.

Cf. A280720.

[n: n in [1..10000000] | IsPrime(n) and IsPrime(5*n+2) and IsPrime(25*n+12) and IsPrime(125*n+62) and IsPrime(625*n+312) and IsPrime(3125*n+1562)] // Vincenzo Librandi, Aug 05 2010

p5Q[n_]:=And@@PrimeQ/@NestList[5#+2&,n,5]
Select[Prime[Range[550000]],p5Q]  (* Harvey P. Dale, Feb 17 2011 *)

a(n) == 31 (mod 42). - John Cerkan, Oct 17 2016

A249573 Smallest prime p that remains prime through exactly n iterations of the function f(x) = 5x + 2.

Original entry on oeis.org

2, 3, 13, 19, 373, 174877, 135859, 18235423, 26588257, 93112729, 376038903103, 7087694466289, 120223669028389

Offset: 0

Felix Fröhlich, Nov 01 2014

Smallest p = prime(i) such that A280720(i) = n, where i is the index of p in A000040. - Felix Fröhlich, Jan 07 2017
From Jon E. Schoenfield, Jan 08 2017: (Start)
a(10) > 10^10.
It seems very likely that a(11) exists. But is it possible that this sequence is finite? Each row of the table below shows, for an interval of width 10^8, the number of primes p within the interval that remain prime through exactly 0 iterations, exactly 1 iteration, etc. E.g., in the interval [10^9, 10^9 + 10^8), there are 4437075 primes p that remain prime through exactly 0 iterations, 326699 that remain prime through exactly 1, 45062 that remain prime through exactly 2, etc.
---------------------------------------------------------------
Fixed interval width = 10^8
---------------------------------------------------------------
Start              Number of successful iterations
of    --------------------------------------------------------
intvl        0      1     2     3    4    5    6    7    8    9
=====  ======= ====== ===== ===== ==== ==== ==== ==== ==== ====
    1  5225638 450798 73434 10139 1308  114   17    4    2    1
10^ 8  4858227 391247 59352  7720  841   84    9    1    1    0
10^ 9  4437075 326699 45062  5438  605   45   10    2    0    0
10^10  4031707 271882 34218  3722  367   30    3    1    0    0
10^11  3689861 228960 26414  2649  251   20    6    0    0    0
10^12  3400459 194999 20675  1973  158   17    1    0    0    0
10^13  3155004 168786 16699  1489  108    6    1    0    0    0
10^14  2940881 147025 13535  1153   81    4    0    0    0    0
10^15  2752985 128743 11275   874   55    5    0    0    0    0
---------------------------------------------------------------
The numbers in column 0 drop at a rate that is not surprising, given that the way that the density of primes drops as numbers get larger. The numbers in the other columns drop more rapidly, in relative terms. Suppose a similar table were constructed using much wider intervals (perhaps with intervals starting not at 1, 10^8, 10^9, 10^10, etc., but at 1, 10^30, 10^31, 10^32, etc.), so that the numbers in, say, column 12 remained positive through several rows, but were dropping by a factor of more than 10 from one row to the next, making it likely that the total number of k-digit primes -- not just those from intervals of a fixed size -- that would remain prime through 12 iterations was actually decreasing as k increased. Would such an outcome suggest that the sequence might be finite? (End)

			With p = 13: 5 * 13 + 2 = 67, 5 * 67 + 2 = 337 and 5 * 337 + 2 = 1687. 67 and 337 are both prime, but 1687 is not, so 13 remains prime through exactly two iterations of 5 * x + 2 and is the smallest prime with this property, so a(2) = 13.

Cf. A005602, A023217, A023313, A023341, A280720.

c[p_] := Block[{k = 1, q = 5*p+2}, While[ PrimeQ[q], q = 5*q+2; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] != n, p = NextPrime@ p]; p]; Array[a, 7] (* Giovanni Resta, Mar 21 2017 *)

for(n=0, 10, forprime(p=2, 1e20, i=0; a=p; while(ispseudoprime(5*a+2), a=5*a+2; i++); if(i==n, print1(p, ", "); break(1))))

a(10) from Charles R Greathouse IV, Jan 13 2017
a(11) from John Cerkan, Mar 20 2017
a(12) from Giovanni Resta, Mar 21 2017

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A023217 Primes p such that 5*p + 2 is also prime.

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A023313 Primes that remain prime through 4 iterations of function f(x) = 5x + 2.

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A023341 Primes that remain prime through 5 iterations of function f(x) = 5x + 2.

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A249573 Smallest prime p that remains prime through exactly n iterations of the function f(x) = 5x + 2.

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