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A278495 a(n) = number of primes in range [n^2, (n+1)^2] that are leaves in "the least squares beanstalk" tree.

%I A278495 #18 Nov 27 2016 22:07:05
%S A278495 1,2,1,2,2,2,1,1,2,4,1,2,1,3,2,4,3,3,3,5,3,2,2,4,4,4,4,3,4,4,4,4,2,3,
%T A278495 3,2,4,2,5,4,6,3,5,4,5,5,4,6,3,3,6,8,4,5,3,5,5,5,4,6,6,7,5,5,7,6,8,8,
%U A278495 8,8,5,5,5,8,7,7,7,3,13,5,8,6,8,7,8,5,14,7,8,8,10,7,5,8,6,7,6,9,4,10,4,9,8,6,8,8,8,6,10,11,13,9
%N A278495 a(n) = number of primes in range [n^2, (n+1)^2] that are leaves in "the least squares beanstalk" tree.
%C A278495 Number of terms of A278494 in range [n^2, (n+1)^2], where A278494 are primes p for which there does not exist any such integer k that k - A002828(k) = p.
%C A278495 In other words, number of primes p in range [n^2, (n+1)^2] for which (A002828(1+p) <> 1) and (A002828(2+p) <> 2) and (A002828(3+p) <> 3) and (A002828(4+p) <> 4).
%C A278495 Conjecture: a(n) > 0 for all n >= 1.
%C A278495 Similar guesses are easy to make but hard to prove. I also conjecture that A277487(n) > 0 for all n > 80, and that both A277486(n) > 0 and A277488(n) > 0 for all n > 7. If any of these claims were proved true, it would imply the proof of Legendre's conjecture as well. See also comments in A014085 and sequences A277888 & A278487.
%H A278495 Antti Karttunen, <a href="/A278495/b278495.txt">Table of n, a(n) for n = 1..10000</a>
%F A278495 For all n >= 1, a(n) <= A014085(n).
%o A278495 (PARI)
%o A278495 istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1
%o A278495 isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
%o A278495 A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From _Charles R Greathouse_ IV, Jul 19 2011
%o A278495 A278495(n) = { my(s = 0); for(k=(n^2),(n+1)^2, if((isprime(k) && (A002828(1+k) <> 1) && (A002828(2+k) <> 2) && (A002828(3+k) <> 3) && (A002828(4+k) <> 4)),s = s+1) ); s; };
%o A278495 for(n=1, 10000, write("b278495.txt", n, " ", A278495(n)));
%o A278495 (Scheme)
%o A278495 (define (A278495 n) (let loop ((k (+ -1 (A000290 (+ 1 n)))) (s 0)) (if (= 1 (A010052 k)) s (loop (- k 1) (+ s (* (A010051 k) (if (zero? (A278216 k)) 1 0)))))))
%Y A278495 Cf. A000290, A002828, A010051, A010052, A014085 (an upper bound), A278216, A278494 (primes that are counted), A278496.
%Y A278495 Cf. also A277486, A277487, A277488.
%K A278495 nonn
%O A278495 1,2
%A A278495 _Antti Karttunen_, Nov 25 2016