A059242 -id:A059242 - cp's OEIS frontend

A133831 Least positive number k != n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

2, 1, 1, 1, 2, 1, 1, 9, 3, 3, 2, 1, 4, 5, 1, 1, 11, 1, 6, 5, 4, 7, 3, 9, 27, 17, 15, 1, 15, 1, 6, 458465, 4, 9, 14, 13, 3, 11, 25, 57, 6, 7, 46, 17, 7, 15, 2, 1009, 30, 23, 6, 21, 2, 33, 1, 1265, 3, 69, 14, 5, 6, 21, 19, 2241, 30, 3, 1, 5, 34, 19, 26, 17, 19, 17, 5, 33, 15, 23, 27

Offset: 1

T. D. Noe, Sep 26 2007

nonn

Does such k exist (so that a(n) is nonzero) for all n? These binary trinomials can also be written as f*2^n+1, where f=2^m+1 for some m, which is reminiscent of the Sierpinski problem (see A076336). Hence if there are no Sierpinski numbers of the form 2^m+1, then a(n) is nonzero for all n.

The PFGW program was used to find a(32), which produces a 138012-digit probable prime. If a(256) is nonzero, it is greater than 10^6.

T. D. Noe, Table of n, a(n) for n = 1..255
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+2^m+1, PRP Top Records.

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).
Closely related problems: A040076 (see also A076336), A067760, A133830 (k < n), A133832 (k > n).
Cf. A095056.

mx=4000; Table[s=1+2^n; k=1; While[k==n || (k

Edited by Peter Munn, Sep 29 2024

A133832 Least number k > n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 3, 5, 13, 6, 7, 9, 9, 18, 19, 14, 13, 15, 17, 17, 81, 20, 19, 30, 33, 26, 27, 38, 81, 27, 35, 31, 33, 35, 31, 42, 458465, 36, 45, 47, 37, 67, 53, 41, 57, 42, 45, 46, 69, 54, 57, 53, 1009, 100, 119, 55, 73, 83, 67, 57, 1265, 74, 69, 66, 113, 75, 101, 66, 2241, 68, 67, 70

Offset: 1

T. D. Noe, Sep 26 2007

nonn

Conjecture: a(n) is nonzero for all n. These binary trinomials can also be written as f*2^n+1, where f=2^m+1 for some m, which is reminiscent of the Sierpinski problem (see A076336). The conjecture is equivalent to no Sierpinski numbers of the form 2^m+1 existing.

The PFGW program was used to find a(32), which produces a 138012-digit probable prime.

T. D. Noe, Table of n, a(n) for n=1..255
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+2^m+1, PRP Top Records.

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).

Cf. A095056, A133830 (k < n equivalent), A133831.

mx=4000; Table[s=1+2^n; k=n+1; While[k

Edited by Peter Munn, Sep 29 2024

A256873 a(n) = 2^(n-1)*(2^n+5).

Original entry on oeis.org

3, 7, 18, 52, 168, 592, 2208, 8512, 33408, 132352, 526848, 2102272, 8398848, 33574912, 134258688, 536952832, 2147647488, 8590262272, 34360393728, 137440264192, 549758435328, 2199028498432, 8796103507968, 35184393060352, 140737530298368, 562950037307392

Offset: 0

M. F. Hasler, Apr 24 2015

For k in A059242, a(k) is in A141548, i.e., A256873 o A059242 is a subsequence of A141548.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

[2^(n-1)*(2^n+5): n in [0..30]]; // Vincenzo Librandi, Apr 24 2015

Table[2^(n - 1) (2^n + 5), {n, 0, 30}] (* Vincenzo Librandi, Apr 24 2015 *)
LinearRecurrence[{6,-8},{3,7},30] (* Harvey P. Dale, Aug 21 2020 *)

PARI
```
A256873(n)=2^(n-1)*(2^n+5)
```

Vec((3-11*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 26 2015

G.f.: (3-11*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 24 2015

a(n) = 6*a(n-1)-8*a(n-2). - Colin Barker, Apr 26 2015

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.

A175173 Primes p such that 2^p + 5 is also prime.

Original entry on oeis.org

3, 5, 11, 47, 53, 191, 34763

Offset: 1

Vincenzo Librandi, Mar 09 2010

nonn

a(8) > 5*10^5. - Robert Price, Sep 18 2015

			For p=3, 2^3+5=13; p=5, 2^5+5=37; p=11, 2^11+5=2053

[p: p in PrimesUpTo(7000) | IsPrime(2^p+5)];

Select[Prime@ Range@ 1000, PrimeQ[2^# + 5] &] (* Michael De Vlieger, Sep 18 2015 *)

A000040 INTERSECT A059242. - R. J. Mathar, May 02 2010

a(7) from Robert Price, Sep 18 2015

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

cp's OEIS Frontend

A133831 Least positive number k != n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

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A133832 Least number k > n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

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Mathematica

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A256873 a(n) = 2^(n-1)*(2^n+5).

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Magma

Mathematica

PARI

PARI

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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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Maple

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A175173 Primes p such that 2^p + 5 is also prime.

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Magma

Mathematica

Formula

Extensions

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