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A232548 Number of ways to write n = p - pi(p) + 2^k + 2^m with 0 < k <= m, where p is an odd prime and pi(p) is the number of primes not exceeding p.

%I A232548 #17 Dec 03 2013 02:44:42
%S A232548 0,0,0,0,1,1,2,1,2,2,3,2,3,3,3,2,3,4,4,3,3,4,4,3,3,5,4,2,3,4,4,3,3,4,
%T A232548 6,4,5,6,4,4,5,4,6,4,4,5,5,4,5,5,4,5,4,5,5,4,3,4,5,5,6,4,4,5,4,5,6,7,
%U A232548 7,4,5,5,6,4,7,6,6,6,4,6,4,7,8,7,6,6,6,6,5,4,9,7,5,4,4,7,6,3,7,8
%N A232548 Number of ways to write n = p - pi(p) + 2^k + 2^m with 0 < k <= m, where p is an odd prime and pi(p) is the number of primes not exceeding p.
%C A232548 Conjecture: a(n) > 0 for all n > 4.
%C A232548 In contrast, R. Crocker proved that there are infinitely many positive odd integers not of the form p + 2^k + 2^m, where p is a prime, and k and m are positive integers.
%C A232548 Qing-Hu Hou has checked the conjecture for n up to 10^7, and found one counterexample: n = 1897048.
%H A232548 Zhi-Wei Sun, <a href="/A232548/b232548.txt">Table of n, a(n) for n = 1..10000</a>
%H A232548 R. Crocker, <a href="http://msp.org/pjm/1971/36-1/p09.xhtml">On the sum of a prime and two powers of two</a>, Pacific J. Math. 36(1971), 103-107.
%H A232548 Z.-W. Sun and M.-H. Le, <a href="http://journals.impan.gov.pl/aa/Inf/99-2-5.html">Integers not of the form c*(2^a + 2^b) + p^{alpha}</a>, Acta Arith. 99(2001), 183-190.
%e A232548 a(7) = 2 since 7 = 3 - pi(3) + 2 + 2^2 = 7 - pi(7) + 2 + 2, with 3 and 7 odd primes.
%e A232548 a(8) = 1 since 8 = 5 - pi(5) + 2 + 2^2 with 5 an odd prime.
%t A232548 a[n_]:=Sum[If[n==Prime[k]-k+2^i+2^j,1,0],{k,2,PrimePi[2n]},{j,1,Log[2,n]},{i,1,j}]
%t A232548 Table[a[n],{n,1,100}]
%o A232548 (PARI) a(n)=my(s,ppi=1); forprime(p=3,, if(p-ppi++>n-4,return(s)); if((n-p+ppi)%2==0 && hammingweight(n-p+ppi)<3,s++)) \\ _Charles R Greathouse IV_, Nov 27 2013
%Y A232548 Cf. A000040, A000079, A000720, A156695, A232398, A232443, A232463.
%K A232548 nonn
%O A232548 1,7
%A A232548 _Zhi-Wei Sun_, Nov 25 2013