A296807 -id:A296807 - cp's OEIS frontend

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.

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.

A297928 a(n) = 24^n + 32^n - 1.

Original entry on oeis.org

4, 13, 43, 151, 559, 2143, 8383, 33151, 131839, 525823, 2100223, 8394751, 33566719, 134242303, 536920063, 2147581951, 8590131199, 34360131583, 137439739903, 549757386751, 2199026401279, 8796099313663, 35184384671743, 140737513521151, 562950003752959, 2251799914348543

Offset: 0

Iain Fox, Jan 08 2018

For n > 0, in binary, this is a 1 followed by n-1 0's followed by 10 followed by n 1's.

			a(0) = 2*4^0 + 3*2^0 - 1 = 4;   in binary, 100.
a(1) = 2*4^1 + 3*2^1 - 1 = 13;  in binary, 1101.
a(2) = 2*4^2 + 3*2^2 - 1 = 43;  in binary, 101011.
a(3) = 2*4^3 + 3*2^3 - 1 = 151; in binary, 10010111.
a(4) = 2*4^4 + 3*2^4 - 1 = 559; in binary, 1000101111.
...

Index entries for linear recurrences with constant coefficients, signature (7,-14,8)

A lower bound for A296807.

Cf. A000918, A085601.

Table[2 4^n+3 2^n-1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{4,13,43},30] (* Harvey P. Dale, Apr 22 2018 *)

PARI
```
a(n) = 2*4^n + 3*2^n - 1
```

first(n) = Vec((4 - 15*x + 8*x^2)/((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^n))

G.f.: (4 - 15*x + 8*x^2)/((1 - x)*(1 - 2*x)*(1 - 4*x)).
E.g.f.: 2*e^(4*x) + 3*e^(2*x) - e^x.
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3), n > 2.
a(n) = A000918(n) + A085601(n).

A297962 Take a prime, convert it to base 2. Remove its most significant digit and its least significant digit; repeat this process. a(n) is the least prime that, in the first n steps of this process, generates n primes.

Original entry on oeis.org

13, 59, 631, 7039, 64063, 761087, 3619327, 74347519, 1577707519, 22200700927, 1668463173631, 290703062134783, 3413184213843967, 121597545150218239

Offset: 1

Paolo Iachia, Paolo P. Lava, Jan 09 2018

From Jon E. Schoenfield, Jan 11 2018: (Start)
a(15) <= 6937869050647642111;
a(16) <= 7088202090368908328959;
a(17) <= 348624306087955627410980863. (End)

			a(1) = 13, because 13 in base 2 is 1101, 10 is the prime 2; and 13 is the least prime with this property.
a(2) = 59, because 59 = 111011_2, 1101_2 is the prime 13, 10_2 is the prime 2; and 59 is the least prime with this property.
a(3) = 631, because 631 = 1001110111_2, 111011_2 is the prime 59, 1101_2 is the prime 13, 10_2 is the prime 2; and 631 is the least prime with this property.

Cf. A296806, A296807.

private static BigInteger SearchAn() { BigInteger BIW, BICore; int tN=10; int j3; static final BigInteger BILim = new BigInteger("1000000000"); static final BigInteger BI2 = new BigInteger("2");for (BIW=BI2; BIW.compareTo(BILim)<0; BIW=BIW.add(BI2)) { BICore = BIW;  for (j3=1;j3

with(numtheory): P:=proc(q) local a,i,k,n,ok,x; x:=1;
for n from 1 to q do for k from x to q do
a:=convert(ithprime(k),binary,decimal);
ok:=1; for i from 1 to n do a:=trunc(a/10) mod 10^(ilog10(a)-1);
if not isprime(convert(a,decimal,binary)) then ok:=0; break; fi; od;
if ok=1 then x:=k; print(ithprime(k)); break; fi; od; od; end: P(10^10);

With[{s = Map[LengthWhile[#, PrimeQ] &@ NestWhileList[((# - 2^Floor@ Log2@ #) - 1)/2 &, #, # > 2 &] &, Prime@ Range[2^18]]}, Map[Prime@ First@ FirstPosition[s, #] &, Range@ Max@ s]] (* Michael De Vlieger, Jan 10 2018 *)

Let C(p) be the result of removing the MSD and the LSD of a prime p. C(p) = (p - 2^floor(log_2(p)) - 1)/2.

a(10)-a(14) from Jon E. Schoenfield, Jan 11 2018

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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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A297928 a(n) = 2*4^n + 3*2^n - 1.

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A297962 Take a prime, convert it to base 2. Remove its most significant digit and its least significant digit; repeat this process. a(n) is the least prime that, in the first n steps of this process, generates n primes.

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A297928 a(n) = 24^n + 32^n - 1.