cp's OEIS Frontend

A262980 Number of ordered ways to write n as p + 2^k + pi(2^m), where p is prime, and k and m are nonnegative integers, and pi(x) denotes the number of primes not exceeding x.

%I A262980 #24 Sep 26 2019 05:50:56
%S A262980 0,0,1,3,4,5,6,7,7,8,8,7,9,7,12,7,10,7,12,9,14,11,12,10,15,8,13,6,12,
%T A262980 7,12,9,13,9,14,11,15,11,18,9,14,8,14,10,18,13,11,9,18,13,17,10,13,7,
%U A262980 15,12,14,10,10,10,15,12,19,11,15,12,16,10,20,12,13,12,20,12,23
%N A262980 Number of ordered ways to write n as p + 2^k + pi(2^m), where p is prime, and k and m are nonnegative integers, and pi(x) denotes the number of primes not exceeding x.
%C A262980 Conjecture: a(n) > 0 for all n > 2.
%C A262980 We have verified this for n up to 2*10^8.
%C A262980 In contrast with this conjecture, in 1971 R. Crocker proved that there are infinitely many positive odd numbers not of the form p + 2^k + 2^m, where p is prime, and k and m are positive integers.
%H A262980 Zhi-Wei Sun, <a href="/A262980/b262980.txt">Table of n, a(n) for n = 1..10000</a>
%H A262980 Roger Crocker, <a href="http://projecteuclid.org/euclid.pjm/1102971271">On the sum of a prime and of two powers of two</a>, Pacific J. Math. 36 (1971), 103-107.
%p A262980 a(3) = 1 since 3 = 2 + 2^0 + pi(2^0) with 2 prime.
%p A262980 a(4) = 3 since 4 = 2 + 2^0 + pi(2) = 2 + 2 + pi(2^0) = 3 + 2^0 + pi(2^0) with 2 and 3 both prime.
%t A262980 f[n_]:=PrimePi[2^n]
%t A262980 Do[r=0;Do[If[f[x]>=n,Goto[aa]];Do[If[PrimeQ[n-f[x]-2^y],r=r+1],{y,0,Log[2,n-f[x]]}];Continue,{x,0,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]
%Y A262980 Cf. A000040, A000079, A000720, A007053, A156695, A232398, A262976.
%K A262980 nonn
%O A262980 1,4
%A A262980 _Zhi-Wei Sun_, Oct 06 2015