A118958 -id:A118958 - cp's OEIS frontend

A065381 Primes not of the form p + 2^k, p prime and k >= 0.

2, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 907, 977, 997, 1019, 1087, 1259, 1549, 1597, 1619, 1657, 1759, 1777, 1783, 1867, 1973, 2203, 2213, 2293, 2377, 2503, 2579, 2683, 2789, 2843, 2879, 2909, 2999, 3119, 3163, 3181, 3187, 3299

Offset: 1

Reinhard Zumkeller, Nov 03 2001

nonn

Sequence is infinite. For example, Pollack shows that numbers which are 1260327937 mod 2863311360 are not of the form p + 2^k for any prime p and k >= 0, and there are infinitely many primes in this congruence class by Dirichlet's theorem. - Charles R Greathouse IV, Jul 20 2014

			127 is a prime, 127-2^0 through 127-2^6 are all nonprimes.

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff. [?Broken link]
P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff.
Lei Zhou, Between 2^n and primes.

Equals A000040 minus A065380.
Cf. A010051, A006285, A102630, A094076, A156695.
Cf. A098237.

a065381 n = a065381_list !! (n-1)
a065381_list = filter f a000040_list where
   f p = all ((== 0) . a010051 . (p -)) $ takeWhile (<= p) a000079_list
-- Reinhard Zumkeller, Nov 24 2011

fQ[n_] := Block[{k = Floor[Log[2, n]], p = n}, While[k > -1 && ! PrimeQ[p - 2^k], k--]; If[k > 0, True, False]]; Drop[Select[Prime[Range[536]], ! fQ[#] &], {2}] (* Robert G. Wilson v, Feb 10 2005; corrected by Arkadiusz Wesolowski, May 05 2012 *)

is(p)=my(k=1);while(kp,return(isprime(p)));0 \\ Charles R Greathouse IV, Jul 20 2014

A078687(A049084(a(n))) = 0; subsequence of A118958. - Reinhard Zumkeller, May 07 2006

Link and cross-reference fixed by Charles R Greathouse IV, Nov 09 2008

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

Original entry on oeis.org

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

A118957 Numbers of the form 2^k + p, where p is a prime less than 2^k.

Original entry on oeis.org

6, 7, 10, 11, 13, 15, 18, 19, 21, 23, 27, 29, 34, 35, 37, 39, 43, 45, 49, 51, 55, 61, 63, 66, 67, 69, 71, 75, 77, 81, 83, 87, 93, 95, 101, 105, 107, 111, 117, 123, 125, 130, 131, 133, 135, 139, 141, 145, 147, 151, 157, 159, 165, 169, 171, 175, 181, 187, 189, 195, 199

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

Complement of A118956; subsequence of A118955.

Cf. A118952, A118958, A156695.

isA118957 := proc(n)
    local twok,p ;
    twok := 1 ;
    while twok < n-1 do
        p := n-twok ;
        if isprime(p) and p < twok then
            return true;
        end if;
        twok := twok*2 ;
    end do:
    return false;
end proc:
for n from 1 to 200 do
    if isA118957(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 27 2015

okQ[n_] := Module[{k, p}, For[k = Ceiling[Log[2, n]], k>1, k--, p = n-2^k; If[2 <= p < 2^k && PrimeQ[p], Return[True]]]; False]; Select[Range[200], okQ] (* Jean-François Alcover, Mar 11 2019 *)

is(n)=isprime(n-2^logint(n,2)) \\ Charles R Greathouse IV, Sep 01 2015; edited Jan 24 2024

from sympy import primepi
def A118957(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(min(x-(m:=1<Chai Wah Wu, Feb 23 2025

A118952(a(n)) = 1.

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)).

cp's OEIS Frontend

A065381 Primes not of the form p + 2^k, p prime and k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A118957 Numbers of the form 2^k + p, where p is a prime less than 2^k.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments