A023242 -id:A023242 - cp's OEIS frontend

A176223 Natural numbers k which give a prime by the function f(k) = 2 * k + 13 for at least two iterations.

2, 5, 8, 17, 23, 35, 38, 47, 50, 68, 77, 80, 107, 110, 113, 140, 152, 170, 218, 227, 233, 245, 248, 278, 287, 317, 320, 332, 353, 365, 380, 392, 407, 437, 458, 467, 485, 500, 518, 542, 575, 590, 602, 605, 623, 635, 638, 710, 740, 743, 770, 803, 827, 842

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 12 2010

nonn

n, p = f(k) = 2 * k + 13, q = f(f(k)) = 4 * k + 39; p and q to be primes.
List of (k,p,q):
(2,17,47) (5,23,59) (8,29,71) (17,47,107) (23,59,131)
(35,83,179) (38,89,191) (47,107,227) (50,113,239) (68,149,311)
(77,167,347) (80,173,359) (107,227,467) (110,233,479) (113,239,491)
(140,293,599) (152,317,647) (170,353,719) (218,449,911) (227,467,947)
(233,479,971) (245,503,1019) (248,509,1031) (278,569,1151) (287,587,1187)
(317,647,1307) (320,653,1319) (332,677,1367) (353,719,1451) (365,743,1499)

			2 * 2 + 13 = 17 = prime(7), 4 * 2 + 39 = 47 = prime(15), 2 is first term.
2 * 5 + 13 = 23 = prime(9), 4 * 5 + 39 = 59 = prime(17), 5 is 2nd term.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A023242, A023272.

k13Q[n_]:=AllTrue[Rest[NestList[2#+13&,n,2]],PrimeQ]; Select[Range[ 1000],k13Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 20 2020 *)

isok(n) = isprime(p=2*n+13) && isprime(2*p+13) \\ Michel Marcus, Jun 28 2013

More terms from Michel Marcus, Jun 28 2013

A176247 Primes p which give a prime iterated by f(p) = 2*p + 13 for at least two steps.

Original entry on oeis.org

2, 5, 17, 23, 47, 107, 113, 227, 233, 317, 353, 467, 743, 827, 1013, 1163, 1223, 1283, 1493, 1697, 1823, 1877, 2063, 2333, 2543, 2957, 3323, 3467, 3767, 3797, 4013, 4397, 4523, 5297, 5393, 5507, 5693, 5717, 5897, 5927, 6053, 6317, 6473, 6737, 6947, 6977

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 13 2010

nonn

Subsequence of A176223.
p, f(p) = 2*p + 13, q = f(f(p)) = 4*p + 39 to be primes.
Necessarily for such primes p > 5, the LSD (least significant digit) is either 3 or 7, since an LSD of 1 gives the LSD of f(p) equal to 5 and an LSD of 9 gives the LSD of f(f(p)) equal to 5.

			f(2) = 17 = prime(7), f(17) = 47 = prime(15), 2 is first term.
f(5) = 23 = prime(9), f(23) = 59 = prime(17), 5 is 2nd term.
Note first resulting palindromic prime: f(3323) = 6659 = prime(858), q = 13331 = prime(1583) = palprime(29).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A023242, A023272, A176223.

Select[Prime@ Range[10^3], AllTrue[NestList[2 # + 13 &, #, 2], PrimeQ] &] (* Michael De Vlieger, Mar 14 2020 *)

isok(n) = isprime(n) && isprime(p=2*n+13) && isprime(2*p+13) \\ Michel Marcus, Jun 28 2013

More terms from Michel Marcus, Jun 28 2013

A176619 Primes p such that 2p + 3, 4p + 9, 3p + 2 and 9p + 8 are also primes.

Original entry on oeis.org

5, 7, 97, 167, 397, 607, 2617, 2707, 7687, 12097, 14407, 16787, 19577, 22307, 23827, 24967, 25717, 28547, 31687, 43037, 43517, 46817, 58967, 59617, 63607, 70237, 70957, 78517, 85027, 96797

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2010

nonn

These primes stay prime under two iterations of p->2p+3 as well as under two iterations of p->3p+2.

For all entries >5 the least significant digit is 7.

			2*5 + 3 = 13 = prime(6),
4*5 + 9 = 29 = prime(10),
3*5 + 2 = 17 = prime(7),
9*5 + 8 = 53 = prime(16); 5 = prime(3) = a(1).

Joe Buhler: Algorithmic Number Theory: Third International Symposium, ANTS-III, Springer New York, 1998
F. Ischebeck: Einladung zur Zahlentheorie, B. I. Wissenschaftsverlag, Mannheim-Leipzig-Wien-Zuerich, 1992

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A005602, A023242, A023246, A059762.

[p: p in PrimesUpTo(100000)|IsPrime(2*p+3) and IsPrime(4*p+9) and IsPrime(3*p+2) and IsPrime(9*p+8 )] // Vincenzo Librandi, Jan 29 2011

Select[Prime[Range[10000]],AllTrue[{2#+3,4#+9,3#+2,9#+8},PrimeQ]&] (* Harvey P. Dale, Dec 15 2024 *)

A023242 INTERSECT A023246.

A247010 Primes p such that (p-3)/2 and 2*p+3 are both prime.

Original entry on oeis.org

7, 13, 17, 29, 89, 97, 137, 197, 229, 277, 337, 349, 397, 557, 617, 797, 929, 937, 1117, 1217, 1237, 1777, 2129, 2309, 2437, 2477, 2617, 2749, 2857, 2909, 3049, 3109, 3137, 3329, 3389, 4057, 4229, 4289, 4409, 5237, 5297, 5417, 5557, 5717, 5857, 6689

Offset: 1

Vincenzo Librandi, Sep 09 2014

A023204 INTERSECT A089531. After 7, all terms are obviously in A002144.

Conjecture: the sequence is infinite.

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

Cf. A023204, A023242, A089531.

[p: p in PrimesUpTo(7000) | IsPrime((p-3)div 2) and IsPrime(2*p+3)];

Select[Prime[Range[900]], And@@PrimeQ/@{(# - 3)/2, 2 # + 3} &]

is(n)=isprime(n) && isprime(2*n+3) && isprime((n-3)\2) \\ Charles R Greathouse IV, Sep 09 2014

def t(i): return 2*i+3
[t(p) for p in primes(5000) if is_prime(t(p)) and is_prime(t(t(p)))] # Bruno Berselli, Sep 09 2014

a(n) = 2*A023242(n) + 3. [Bruno Berselli, Sep 09 2014]

A244086 Primes p such that p remains prime through 7 iterations of function f(x) = 2*x + 3.

Original entry on oeis.org

477727, 507757, 30596497, 33145687, 36180527, 61192997, 141339217, 148590307, 193394347, 297180617, 374066267, 395534747, 398001547, 419795137, 488716897

Offset: 1

Vincenzo Librandi, Jun 24 2014

Cf. similar sequences with k iterations: A067076 (k=1), A023242 (k=2), A023273 (k=3), A023303 (k=4), A023331 (k=5), A175160 (k=6), this sequence (k=7).

[p: p in PrimesUpTo(600000000) | forall{i: i in [1..7] | IsPrime(2^i*p+3*(2^i-1))}];

Select[Prime[Range[600000]], And@@PrimeQ[NestList[2 # + 3 &, #, 7]] &]

Inappropriate MAGMA code simplified and other terms added from Bruno Berselli, Jun 24 2014

cp's OEIS Frontend

A176223 Natural numbers k which give a prime by the function f(k) = 2 * k + 13 for at least two iterations.

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A176247 Primes p which give a prime iterated by f(p) = 2*p + 13 for at least two steps.

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A176619 Primes p such that 2p + 3, 4p + 9, 3p + 2 and 9p + 8 are also primes.

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A247010 Primes p such that (p-3)/2 and 2*p+3 are both prime.

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A244086 Primes p such that p remains prime through 7 iterations of function f(x) = 2*x + 3.

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Magma

Mathematica

Extensions