A118952 -id:A118952 - cp's OEIS frontend

A109925 Number of primes of the form n - 2^k.

0, 0, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 0, 1, 2, 3, 1, 4, 0, 2, 1, 2, 0, 3, 0, 1, 1, 2, 1, 3, 1, 3, 0, 2, 1, 4, 0, 1, 1, 2, 1, 5, 0, 2, 1, 3, 0, 3, 0, 1, 1, 3, 0, 2, 0, 1, 1, 3, 1, 4, 0, 1, 1, 2, 1, 5, 0, 2, 1, 2, 1, 6, 0, 3, 0, 2, 1, 3, 0, 3, 1, 2, 0, 4, 0, 1, 1, 3, 0, 3, 0, 2, 0, 1, 1, 3, 0, 2, 1, 2, 1, 6

Offset: 1

Amarnath Murthy, Jul 17 2005

Erdős conjectures that the numbers in A039669 are the only n for which n-2^r is prime for all 2^rT. D. Noe and Robert G. Wilson v, Jul 19 2005

a(A006285(n)) = 0. - Reinhard Zumkeller, May 27 2015

			a(21) = 4, 21-2 =19, 21-4 = 17, 21-8 = 13, 21-16 = 5, four primes.
127 is the smallest odd number > 1 such that a(n) = 0: A006285(2) = 127. - _Reinhard Zumkeller_, May 27 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas Bloom, Let f(n) count the number of solutions to n = p + 2^k for prime p and k >= 0. Is it true that f(n) = o(log n)?, Erdős Problems.
Terence Tao, Erdős problem database, see no. 236.

Cf. A039669, A109926, A175956, A156695.
Cf. A000079, A000040, A010051, A006285.
Cf. A118954, A118955, A118952, A078687.

a109925 n = sum $ map (a010051' . (n -)) $ takeWhile (< n)  a000079_list
-- Reinhard Zumkeller, May 27 2015

a109925:=function(n); count:=0; e:=1; while e le n do if IsPrime(n-e) then count+:=1; end if; e*:=2; end while; return count; end function; [ a109925(n): n in [1..105] ]; // Klaus Brockhaus, Oct 30 2010

A109925 := proc(n)
    a := 0 ;
    for k from 0 do
        if n-2^k < 2 then
            return a ;
        elif isprime(n-2^k) then
            a := a+1 ;
        end if;
    end do:
end proc:
seq(A109925(n),n=1..80) ; # R. J. Mathar, Mar 07 2022

Table[cnt=0; r=1; While[rRobert G. Wilson v, Jul 21 2005 *)
Table[Count[n - 2^Range[0, Floor[Log2[n]]], ?PrimeQ], {n, 110}] (* _Harvey P. Dale, Oct 21 2024 *)

a(n)=sum(k=0,log(n)\log(2),isprime(n-2^k)) \\ Charles R Greathouse IV, Feb 19 2013

from sympy import isprime
def A109925(n): return sum(1 for i in range(n.bit_length()) if isprime(n-(1<Chai Wah Wu, Nov 29 2023

a(A118954(n))=0, a(A118955(n))>0; A118952(n)<=a(n); A078687(n)=a(A000040(n)). - Reinhard Zumkeller, May 07 2006

G.f.: ( Sum_{i>=0} x^(2^i) ) * ( Sum_{j>=1} x^prime(j) ). - Ilya Gutkovskiy, Feb 10 2022

Corrected and extended by T. D. Noe and Robert G. Wilson v, Jul 19 2005

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)).

A118956 Numbers that cannot be written as 2^k + p with p prime < 2^k.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 14, 16, 17, 20, 22, 24, 25, 26, 28, 30, 31, 32, 33, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 53, 54, 56, 57, 58, 59, 60, 62, 64, 65, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 86, 88, 89, 90, 91, 92, 94, 96, 97, 98, 99, 100, 102, 103, 104, 106

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

Complement of A118957.

A118954 is a subsequence.

Cf. A118952, A118954, A118957.

nn=15;Complement[Range[nn^2],Flatten[Table[c=2^n;c+Prime[ Range[ PrimePi[ c]]],{n,2,nn}]]] (* Harvey P. Dale, Sep 14 2012 *)

from sympy import primepi
def A118956(n):
    def f(x): return int(n+sum(primepi(min(x-(m:=1<Chai Wah Wu, Feb 23 2025

A118952(a(n)) = 0.

cp's OEIS Frontend

A109925 Number of primes of the form n - 2^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

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

A118956 Numbers that cannot be written as 2^k + p with p prime < 2^k.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula