A109925 - OEIS

%I #41 Oct 21 2024 12:29:07

%S 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,

%T 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,

%U 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

%N Number of primes of the form n - 2^k.

%C 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

%C a(A006285(n)) = 0. - _Reinhard Zumkeller_, May 27 2015

%H T. D. Noe, <a href="/A109925/b109925.txt">Table of n, a(n) for n = 1..10000</a>

%F a(A118954(n))=0, a(A118955(n))>0; A118952(n)<=a(n); A078687(n)=a(A000040(n)). - _Reinhard Zumkeller_, May 07 2006

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

%e a(21) = 4, 21-2 =19, 21-4 = 17, 21-8 = 13, 21-16 = 5, four primes.

%e 127 is the smallest odd number > 1 such that a(n) = 0: A006285(2) = 127. - _Reinhard Zumkeller_, May 27 2015

%p A109925 := proc(n)

%p     a := 0 ;

%p     for k from 0 do

%p         if n-2^k < 2 then

%p             return a ;

%p         elif isprime(n-2^k) then

%p             a := a+1 ;

%p         end if;

%p     end do:

%p end proc:

%p seq(A109925(n),n=1..80) ; # _R. J. Mathar_, Mar 07 2022

%t Table[cnt=0; r=1; While[r<n, If[PrimeQ[n-r], cnt++ ]; r=2r]; cnt, {n, 150}] (Noe)

%t 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 *)

%t Table[Count[n - 2^Range[0, Floor[Log2[n]]], _?PrimeQ], {n, 110}] (* _Harvey P. Dale_, Oct 21 2024 *)

%o (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

%o (PARI) a(n)=sum(k=0,log(n)\log(2),isprime(n-2^k)) \\ _Charles R Greathouse IV_, Feb 19 2013

%o (Haskell)

%o a109925 n = sum $ map (a010051' . (n -)) $ takeWhile (< n)  a000079_list

%o -- _Reinhard Zumkeller_, May 27 2015

%o (Python)

%o from sympy import isprime

%o def A109925(n): return sum(1 for i in range(n.bit_length()) if isprime(n-(1<<i))) # _Chai Wah Wu_, Nov 29 2023

%Y Cf. A039669, A109926, A175956, A156695.

%Y Cf. A000079, A000040, A010051, A006285.

%Y Cf. A118954, A118955, A118952, A078687.

%K easy,nonn

%O 1,4

%A _Amarnath Murthy_, Jul 17 2005

%E Corrected and extended by _T. D. Noe_ and _Robert G. Wilson v_, Jul 19 2005