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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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)

LINKS

Iain Fox, Table of n, a(n) for n = 1..10000

Ken Abbott, Prime Cores., Number Theory group on LinkedIn.com

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

Sequence in context: A228324 A171122 A342253 * A143788 A165459 A108794

Adjacent sequences:  A296803 A296804 A296805 * A296807 A296808 A296809

KEYWORD

nonn,base,easy

AUTHOR

Paolo P. Lava, Paolo Iachia, Dec 21 2017

STATUS

approved

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Last modified May 15 18:26 EDT 2021. Contains 343920 sequences. (Running on oeis4.)