A057195 -id:A057195 - cp's OEIS frontend

A217349 Numbers k such that 4^k + 7 is prime.

1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 19, 22, 39, 44, 49, 63, 80, 87, 102, 107, 294, 305, 399, 463, 595, 599, 903, 944, 1324, 1727, 1755, 1932, 1935, 4485, 6165, 6665, 9438, 11169, 19859, 27503, 55392, 86235, 98217, 117855, 123640, 134204, 139660, 150437, 157634, 186475, 236129, 283248, 390142, 410178

Offset: 1

Vincenzo Librandi, Oct 01 2012

nonn

The next terms are > 4.1*10^5. - Elmo R. Oliveira, Nov 29 2023

			For k = 14, 4^14 + 7 = 268435463 is prime.

Cf. A057195, A059266, A089437, A104066 (associated primes).

Select[Range[0, 5000], PrimeQ[4^# + 7] &]

is(n)=ispseudoprime(4^n+7) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = A057195(n)/2.

Extended using A057195 terms by Michel Marcus, Aug 28 2015

a(51)-a(54) derived from A057195 by Elmo R. Oliveira, Nov 29 2023

A296806 Take a prime, convert it to base 2, remove its most significant digit and its least significant digit and convert it back to base 10. Sequence lists primes that generate another prime by this process.

Original entry on oeis.org

13, 23, 31, 37, 43, 47, 59, 71, 79, 103, 127, 139, 151, 163, 167, 191, 211, 223, 251, 263, 271, 283, 331, 379, 463, 523, 547, 571, 587, 599, 607, 619, 631, 647, 659, 691, 719, 727, 739, 787, 811, 827, 839, 859, 907, 911, 967, 971, 991, 1031, 1039, 1051, 1063, 1087

Offset: 1

Paolo P. Lava, Paolo Iachia, Dec 21 2017

From an idea of Ken Abbott (see link).
From Paolo Iachia, Dec 21 2017: (Start)
Let us call these numbers "core of a prime".
Let C(q) be the core of a prime q.
Then C(q) = (q - 2^floor(log_2(q)) - 1)/2.
Examples: C(59) = (59 - 2^5 - 1)/2 = 13; C(71) = (71 - 2^6 - 1)/2 = 3; C(73) = (73 - 2^6 - 1)/2 = 4; C(251) = (251 - 2^7 - 1)/2 = 61.
0 <= C(q) <= 2^(floor(log_2(q)) - 1) - 1. The minimum (0) occurs when q = 2^n+1, with n > 2. Example: 17 = 2^4+1, C(17) = (17 - 2^4 - 1)/2 = 0.  The maximum is reached when q = 2^n-1 is a Mersenne prime. Example: 127 = 2^7 - 1, C(127) = (127 - 2^6 - 1)/2 = 31 = 2^5 - 1.
The last example is particularly interesting, as both the prime q and its core are Mersenne primes. The same holds for C(31) = 7 and for C(524247) = 131071, with 524247 = 2^19-1 and 131071 = 2^17-1, both Mersenne primes. Are there more such cases?
Note that the core of Mersenne number (prime or not) is a Mersenne number by definition. Counterexamples include C(8191) = 2047, with 8191 = 2^13 - 1, a Mersenne prime, but 2047 = 2^11 - 1 = 23*89, a Mersenne number not prime, and C(131071) = 32767 = 2^15 - 1 = 7*31*151, with 2 of its factors being Mersenne primes.
Primes whose binary expansion is of the form q = 1 0 ... 0 c_1 c_2 ... c_k 1 - with none or any number of consecutive 0's and with binary core c_1 c_2 ... c_k, k >= 0 - share the same core value. Let p = C(q), then we can write, in decimal form, q = (2p+1) + 2^n, for an appropriate n. While the property is true for p prime, it can be generalized to any positive integer.
Conjecture: for any positive integer p, there are infinitely many primes q for which there exists an integer n such that q-(2p+1) = 2^n. (End)

			13 in base 2 is 1101 and 10 is 2;
23 in base 2 is 10111 and 011 is 3;
31 in base 2 is 11111 and 111 is 7.

Iain Fox, Table of n, a(n) for n = 1..10000
Ken Abbott, Prime Cores, Number Theory group on LinkedIn.

Cf. A000030, A000225, A001348, A004676, A010879, A057195, A057196, A059242, A123250, A296807.

with(numtheory): P:=proc(q) local a,b,c,j,n,ok,x;  x:=5; for n from x to q do ok:=1; a:=convert(ithprime(n),base,2); b:=nops(a)-1; while a[b]=0 do b:=b-1; od; c:=0;
for j from b by -1 to 2 do c:=2*c+a[j]; od;if isprime(c) then x:=n; print(ithprime(n)); fi; od; end: P(10^6);
# simpler alternative:
select(t -> isprime(t) and isprime((t - 2^ilog2(t) - 1)/2), [seq(i,i=3..10^4,2)]); # Robert Israel, Dec 28 2017

Select[Prime[Range[200]],PrimeQ[FromDigits[Most[Rest[IntegerDigits[ #,2]]],2]]&] (* Harvey P. Dale, Jul 19 2020 *)

lista(nn) = forprime(p=13, nn, if(isprime((p - 2^logint(p, 2) - 1)/2), print1(p, ", "))) \\ Iain Fox, Dec 28 2017

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    p = 7
    while True:
        if isprime(int(bin(p)[3:-1], 2)):
            yield p
        p = nextprime(p)
print(list(islice(agen(), 54))) # Michael S. Branicky, May 16 2022

Primes q such that C(q) = (q - 2^floor(log_2(q)) - 1)/2 is prime too.

A217381 Numbers k such that 8^k + 7 is prime.

Original entry on oeis.org

2, 6, 10, 26, 42, 58, 68, 196, 266, 602, 1170, 1288, 1290, 2990, 4110, 6292, 7446, 36928, 57490, 65478, 78570, 188832, 273452

Offset: 1

Vincenzo Librandi, Oct 02 2012

All terms are equal to 1/3 of the multiples of 3 in A057195.

Naturally these numbers are even because (9-1)^(2n+1)+7 is divisible by 3. - Bruno Berselli, Oct 03 2012

Cf. A057195, A217354, A217355.

Cf. A144360 (associated primes).

Mathematica
```
Select[Range[10000], PrimeQ[8^# + 7] &]
```

is(n)=ispseudoprime(8^n+7) \\ Charles R Greathouse IV, Jun 13 2017

a(18)-a(22) from A057195 by Robert Price, Jul 23 2017

a(23) from the data at A057195 added by Amiram Eldar, Jul 23 2025

A155779 Primes p such that 2^(p-1)+7 is prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 29, 31, 79, 89, 127, 1889, 3511, 8971, 13331

Offset: 1

Vincenzo Librandi, Jan 27 2009

nonn

Primes of the form 1+A057195(k). [R. J. Mathar, Feb 19 2009]

Select[Prime[Range[15000]], PrimeQ[(2^(# - 1) + 7)] &] (* Vincenzo Librandi, Jun 24 2013 *)

A361744 A(n,k) is the least m such that there are k primes in the set {prime(n) + 2^i | 1 <= i <= m}, or -1 if no such number exists; square array A(n,k), n > 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 5, 4, 1, 6, 11, 6, 3, 2, 7, 47, 8, 5, 4, 1, 12, 53, 10, 7, 8, 13, 2, 15, 141, 16, 9, 20, 21, 6, 3, 16, 143, 18, 15, 38, 33, 30, 7, 1, 18, 191, 20, 23, 64, 81, 162, 39, 3, 4, 28, 273, 28, 29, 80, 129, 654, 79, 5, 12, 2

Offset: 2

Jean-Marc Rebert, Mar 22 2023

			p = prime(2) = 3, m=1, u = {p + 2^k | 1 <= k <= m} = {5} contains one prime, and no lesser m satisfies this, so A(2,1) = 1.
Square array A(n,k) n > 1 and k >= 1 begins:
 1,     2,     3,     4,     6,     7,    12,    15,    16,    18, ...
 1,     3,     5,    11,    47,    53,   141,   143,   191,   273, ...
 2,     4,     6,     8,    10,    16,    18,    20,    28,    30, ...
 1,     3,     5,     7,     9,    15,    23,    29,    31,    55, ...
 2,     4,     8,    20,    38,    64,    80,   292,  1132,  4108, ...
 1,    13,    21,    33,    81,   129,   285,   297,   769,  3381, ...
 2,     6,    30,   162,   654,   714,  1370,  1662,  1722,  2810, ...
 3,     7,    39,    79,   359,   451,  1031,  1039, 11311, 30227, ...
 1,     3,     5,     7,     9,    13,    15,    17,    23,    27, ...

Cf. A057732 (1st row), A094076 (1st column).
Cf. A361679.
Cf. A019434 (primes 2^n+1), A057732 (2^n+3), A059242 (2^n+5), A057195 (2^n+7), A057196(2^n+9), A102633 (2^n+11), A102634 (2^n+13), A057197 (2^n+15), A057200 (2^n+17), A057221 (2^n+19), A057201 (2^n+21), A057203 (2^n+23).
Cf. A205558 and A231232 (with 2*k instead of 2^k).

A(n, k)= {my(nb=0, p=prime(n), m=1); while (nb

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A217349 Numbers k such that 4^k + 7 is prime.

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A296806 Take a prime, convert it to base 2, remove its most significant digit and its least significant digit and convert it back to base 10. Sequence lists primes that generate another prime by this process.

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A217381 Numbers k such that 8^k + 7 is prime.

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A155779 Primes p such that 2^(p-1)+7 is prime.

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Mathematica

A361744 A(n,k) is the least m such that there are k primes in the set {prime(n) + 2^i | 1 <= i <= m}, or -1 if no such number exists; square array A(n,k), n > 1, k >= 1, read by antidiagonals.

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PARI