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

1,4

Comments

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

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

Links

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

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

Example

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

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[r<n, If[PrimeQ[n-r], cnt++ ]; r=2r]; cnt, {n, 150}] (Noe)
f[n_] := Count[ PrimeQ[n - 2^Range[0, Floor[ Log[2, n]]]], True]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Jul 21 2005 *)

Prog

(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
(PARI) a(n)=sum(k=0, log(n)\log(2), isprime(n-2^k)) \\ Charles R Greathouse IV, Feb 19 2013
(Haskell)
a109925 n = sum $ map (a010051' . (n -)) $ takeWhile (< n)  a000079_list
-- Reinhard Zumkeller, May 27 2015
(Python)
from sympy import isprime
def A109925(n): return sum(1 for i in range(n.bit_length()) if isprime(n-(1<<i))) # Chai Wah Wu, Nov 29 2023

Crossrefs

Cf. A039669, A109926, A175956, A156695.

Cf. A000079, A000040, A010051, A006285.

Cf. A118954, A118955, A118952, A078687.

Sequence in context: A174314 A237253 A080634 * A306260 A180227 A001468

Adjacent sequences: A109922 A109923 A109924 * A109926 A109927 A109928

Keyword

easy,nonn

Author

Amarnath Murthy, Jul 17 2005

Extensions

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

Status

approved