A091932 -id:A091932 - cp's OEIS frontend

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

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 *)

A065380 Primes of the form p + 2^k, p prime and k >= 0.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Reinhard Zumkeller, Nov 03 2001

nonn

			a(3) = 11 = 3 + 2^3 = 7 + 2^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000079, A065381, A156695.

a065380 n = a065380_list !! (n-1)
a065380_list = filter f a000040_list where
   f p = any ((== 1) . a010051 . (p -)) $ takeWhile (<= p) a000079_list
-- Reinhard Zumkeller, Nov 24 2011

With[{upto=300},Select[Union[Select[Flatten[Outer[Plus,Prime[Range[ PrimePi[upto]]],2^Range[0,Floor[Log[2,upto]]]]],PrimeQ]],#<=upto&]] (* Harvey P. Dale, Feb 28 2012 *)

A078687(A049084(a(n))) > 0; A091932 is a subsequence. - Reinhard Zumkeller, May 07 2006

A118953 Number of ways to write the n-th prime as 2^k + p, where p is prime and p < 2^k.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

0 <= a(n) <= 1, a(n) <= A078687(n);
a(A049084(A118958(n))) = 0, a(A049084(A091932(n))) = 1;
a(n) = A118952(A000040(n)).

A091931 Change the first bit to 0 in binary notation for the n-th prime.

Original entry on oeis.org

0, 1, 1, 3, 3, 5, 1, 3, 7, 13, 15, 5, 9, 11, 15, 21, 27, 29, 3, 7, 9, 15, 19, 25, 33, 37, 39, 43, 45, 49, 63, 3, 9, 11, 21, 23, 29, 35, 39, 45, 51, 53, 63, 65, 69, 71, 83, 95, 99, 101, 105, 111, 113, 123, 1, 7, 13, 15, 21, 25, 27, 37, 51, 55, 57, 61, 75, 81, 91, 93, 97, 103

Offset: 1

Reinhard Zumkeller, Feb 14 2004

a(n) = A053645(A000040(n)).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A091932.

FromDigits[Rest[IntegerDigits[#,2]],2]&/@Prime[Range[80]] (* Harvey P. Dale, Apr 20 2012 *)

for(n=1,72,p=prime(n);p-=2^(#binary(p)-1);print1(p,", ")) \\ Washington Bomfim, Jan 18 2011

a(n) = A000040(n) - 2^(A035100(n)-1).

A152084 Primes p such that p + 2^floor(log_2(p)) is prime.

Original entry on oeis.org

3, 7, 11, 31, 41, 47, 67, 73, 103, 109, 127, 149, 179, 239, 251, 307, 313, 331, 337, 397, 421, 463, 487, 521, 557, 617, 641, 659, 701, 719, 809, 887, 911, 941, 947, 971, 977, 1019, 1039, 1063, 1087, 1117, 1129, 1213, 1249, 1327, 1399, 1423, 1453, 1567, 1597

Offset: 1

Leroy Quet, Nov 23 2008

nonn

a(n) + 2^floor(log_2(a(n))) = A152085(n).
If a(n) is written in binary and the leftmost 1 is replaced with "10", then we would have the binary representation of A152085(n), which is a prime.
Sequence A091932 contains the related primes p where p - 2^floor(log_2(p)) = prime.

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

A152085, A091932

filter:= n -> isprime(n) and isprime(n + 2^ilog2(n)):
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Mar 14 2024

Extended by Ray Chandler, Nov 26 2008

A191235 Primes p such that the binary representation of p is the concatenation of the binary representations of prime 2 and an odd prime.

Original entry on oeis.org

11, 23, 43, 83, 181, 353, 359, 383, 643, 661, 691, 709, 739, 751, 1301, 1307, 1361, 1373, 1433, 1481, 1487, 1511, 1523, 2617, 2647, 2689, 2707, 2731, 2749, 2767, 2791, 2857, 2887, 3001, 3019, 3061, 3067, 5147, 5189, 5297, 5309, 5333, 5387, 5393

Offset: 1

Juri-Stepan Gerasimov, May 27 2011

The odd primes arising in computing the sequence are 3, 7, 11, 19, 53, 97, 103, 127, 131, 149, 179, 197, 227, 239, ...

Primes whose binary representation equals the binary representation of some prime preceded by 10. - Klaus Brockhaus, May 29 2011

			11 is in the sequence because 11, 2, 3 in binary are resp. 1011, 10, 11.
83 is in the sequence because 83, 2, 19 in binary are resp. 1010011, 10, 10011.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A004676, A007088, A091932, A080165, A090423, A053644.

[ p: p in PrimesInInterval(3, 6100) | exists(q){ k: k in PrimesUpTo(p div 3) | Intseq(p, 2) eq Intseq(k, 2) cat [0, 1] } ]; // Klaus Brockhaus, May 29 2011

A053644(n)=my(k=1);while(k<=n,k<<=1);k>>1;
forprime(p=2,1e3,if(isprime(k=4*A053644(p)+p),print1(k", "))) \\ Charles R Greathouse IV, May 27 2011

a(4) corrected, a(15)-a(56) added by Charles R Greathouse IV, May 27 2011

A188677 Primes p such that the minimum value of |p-2^x|, x>0, is also a prime.

Original entry on oeis.org

11, 13, 19, 23, 29, 37, 43, 53, 59, 61, 67, 71, 83, 97, 109, 131, 139, 151, 157, 181, 197, 227, 233, 239, 251, 263, 269, 293, 317, 353, 359, 383, 409, 433, 439, 499, 509, 523, 541, 571, 601, 613, 619, 643, 661, 691, 709, 739, 751, 773, 797, 827, 857

Offset: 1

Benoit Cloitre & Robert G. Wilson v, Apr 08 2011

nonn

Originally submitted by Benoit Cloitre, Dec 17 2002 as A078686 and corrected by Robert G. Wilson v, Apr 08 2011.

Cf. A078686, A086081, A091932.

fQ[n_] := Block[{x = Floor@ Log2@ n}, PrimeQ@ Min[n - 2^x, 2^(x+1) - n]]; Select[ Prime@ Range@ 150, fQ] (* Robert G. Wilson v, Apr 08 2011 *)

is(n)=if(isprime(n),my(x=log(n)\log(2));isprime(min(abs(n-1<Charles R Greathouse IV, Jan 10 2013

Intersection of A086081 and A091932. - Robert G. Wilson v, May 27 2011

cp's OEIS Frontend

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

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A065380 Primes of the form p + 2^k, p prime and k >= 0.

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A118953 Number of ways to write the n-th prime as 2^k + p, where p is prime and p < 2^k.

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A091931 Change the first bit to 0 in binary notation for the n-th prime.

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A152084 Primes p such that p + 2^floor(log_2(p)) is prime.

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Maple

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A191235 Primes p such that the binary representation of p is the concatenation of the binary representations of prime 2 and an odd prime.

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Magma

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A188677 Primes p such that the minimum value of |p-2^x|, x>0, is also a prime.

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Keywords

Comments

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Programs

Mathematica

PARI

Formula