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

Author

Amarnath Murthy, Jul 17 2005

Keywords

• easy
• nonn

Comments

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

Examples

			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
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A039669, A109926, A175956, A156695.

Cf. A000079, A000040, A010051, A006285.

Cf. A118954, A118955, A118952, A078687.

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

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