A118953 -id:A118953 - cp's OEIS frontend

A091932 Primes that remain prime when their leading digit in binary representation is replaced by 0.

7, 11, 13, 19, 23, 29, 37, 43, 61, 67, 71, 83, 101, 107, 131, 139, 151, 157, 181, 199, 211, 229, 241, 263, 269, 293, 317, 353, 359, 383, 419, 449, 467, 479, 523, 541, 571, 601, 613, 619, 643, 661, 691, 709, 739, 751, 769, 823, 829, 859, 991, 1021, 1031, 1061

Offset: 1

Reinhard Zumkeller, Feb 14 2004

nonn

A053645(a(n)) is prime.

Primes p such that p - 2^floor(log_2(p)) is prime - T. D. Noe, Apr 08 2011

			A000040(12)=37 --> '100101' --> '[1]00101' --> '[0]00101' --> '101' --> 5, therefore 37 is a term.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A091931.

Cf. A118958.

Select[Prime[Range[100]], PrimeQ[# - 2^Floor[Log[2, #]]] &] (* T. D. Noe, Apr 08 2011 *)
Select[Prime[Range[200]],PrimeQ[FromDigits[Rest[ IntegerDigits[ #,2]],2]]&] (* Harvey P. Dale, Apr 08 2016 *)

from sympy import isprime, primerange
def ok(p): return isprime((1 << (p.bit_length()-1)) ^ p)
def aupto(lim): return [p for p in primerange(1, lim+1) if ok(p)]
print(aupto(1061)) # Michael S. Branicky, Jul 11 2021

A118953(A049084(a(n))) = 1; subsequence of A065380. - Reinhard Zumkeller, May 07 2006

A078687 Number of x>=0 such that prime(n)-2^x is prime.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 2, 0, 3, 1, 4, 0, 2, 2, 1, 3, 2, 1, 4, 1, 1, 2, 4, 2, 1, 3, 3, 1, 1, 3, 0, 2, 2, 1, 3, 2, 1, 2, 3, 1, 1, 2, 2, 0, 0, 2, 2, 3, 1, 2, 0, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 0, 1, 3, 2, 1, 1, 3, 1, 4

Offset: 1

Benoit Cloitre, Dec 17 2002

nonn

			prime(17)=59 and only 59-2^3 = 53 is prime hence a(17)=1

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A156695.

f[p_] := Block[{c = exp = 0, lmt = 1 + Floor@ Log2@ p}, While[exp < lmt, If[ PrimeQ[p - 2^exp], c++]; exp++]; c]; Array[ f@ Prime@# &, 105] (* Robert G. Wilson v, Jul 07 2014 *)

a(n)=sum(i=0,floor(log(prime(n))/log(2)),if(isprime(prime(n)-2^i),1,0))

a(A049084(A065381(n)))=0, a(A049084(A065380(n)))=1; A118953(n)<=a(n); a(n)=A109925(A000040(n)). - Reinhard Zumkeller, May 07 2006

A118958 Primes that cannot be written as 2^k + p with p prime < 2^k.

Original entry on oeis.org

2, 3, 5, 17, 31, 41, 47, 53, 59, 73, 79, 89, 97, 103, 109, 113, 127, 137, 149, 163, 167, 173, 179, 191, 193, 197, 223, 227, 233, 239, 251, 257, 271, 277, 281, 283, 307, 311, 313, 331, 337, 347, 349, 367, 373, 379, 389, 397, 401, 409, 421, 431, 433, 439, 443

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

A118953(A049084(a(n))) = 0; A065381 is a subsequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A118957, A091932, A156695.

Cf. A049084, A065381, A118953.

filter:= proc(n) not isprime(n-2^ilog2(n)) end proc:
select(filter, [seq(ithprime(i),i=1..100)]); # Robert Israel, Jan 27 2021

okQ[n_] := !PrimeQ[n-2^(Length[IntegerDigits[n, 2]]-1)];
Select[Prime[Range[100]], okQ] (* Jean-François Alcover, Feb 04 2023 *)

A118952 Characteristic function of numbers that can be written as p+2^k, where p is prime and p less than 2^k (A118957).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

0 <= a(n) <= 1, a(n) <= A109925(n);
a(A118956(n)) = 0, a(A118957(n)) = 1;
A118953(n) = a(A000040(n)).

Index entries for characteristic functions

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A091932 Primes that remain prime when their leading digit in binary representation is replaced by 0.

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A078687 Number of x>=0 such that prime(n)-2^x is prime.

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A118958 Primes that cannot be written as 2^k + p with p prime < 2^k.

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A118952 Characteristic function of numbers that can be written as p+2^k, where p is prime and p less than 2^k (A118957).

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