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A296806
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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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3
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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
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OFFSET
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1,1
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COMMENTS
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From an idea of Ken Abbott (see link).
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)
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LINKS
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Ken Abbott, Prime Cores, Number Theory group on LinkedIn.
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FORMULA
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Primes q such that C(q) = (q - 2^floor(log_2(q)) - 1)/2 is prime too.
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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Select[Prime[Range[200]], PrimeQ[FromDigits[Most[Rest[IntegerDigits[ #, 2]]], 2]]&] (* Harvey P. Dale, Jul 19 2020 *)
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PROG
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(PARI) lista(nn) = forprime(p=13, nn, if(isprime((p - 2^logint(p, 2) - 1)/2), print1(p, ", "))) \\ Iain Fox, Dec 28 2017
(Python)
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)
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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