%I #22 Nov 14 2016 10:35:39

%S 1,0,1,0,0,1,2,1,1,0,2,1,2,0,3,2,0,3,0,2,0,1,4,2,3,2,4,2,0,3,3,2,5,3,

%T 4,3,3,3,2,4,2,2,4,3,3,3,6,3,1,3,4,2,6,3,3,2,5,5,5,5,4,3,7,4,4,6,4,2,

%U 4,6,5,5,5,4,7,4,4,7,4,0,5,6,7,4,4,9,4,5,2,6,6,7,11,3,6,4,9,5,7,7,7,6,8,8,7,6,4,6,5,7,8,5,9,8,8,5,12,7,5,6

%N a(n) = number of primes encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

%C Number of primes on row n of A276574, after the initial zero-row.

%C Note how for the most n in range 1..10000, a(n) < A277486(n), even though for the most n in the same range A277890(n) < A277891(n). In range n=1..10000, there are only 209 cases where a(n) >= A277486(n).

%C On the other hand, when a(n) is compared to A277488(n), there is no such marked bias.

%H Antti Karttunen, <a href="/A277487/b277487.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) <= A277891(n).

%e For n=3, starting from k = ((3+1)^2)-1, and iterating k -> A255131(k), yields 15 -> 11 -> 8, where the iteration stops as the next lower number one less than a square has been reached. Of these numbers only 11 is a prime, thus a(3) = 1.

%o (PARI)

%o 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 isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7

%o A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From _Charles R Greathouse_ IV, Jul 19 2011

%o A277487(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + if(isprime(k),1,0); k = k - A002828(k)); s; };

%o for(n=1, 10000, write("b277487.txt", n, " ", A277487(n)));

%o (Scheme)

%o (define (A277487 n) (let ((org_k (- (A000290 (+ 1 n)) 1))) (let loop ((k org_k) (s 0)) (if (and (< k org_k) (= 1 (A010052 (+ 1 k)))) s (loop (- k (A002828 k)) (+ s (A010051 k)))))))

%Y Cf. A000290, A002828, A010052, A276574, A277486, A277488, A277890, A277891, A278167 (partial sums).

%Y Cf. A277888.

%K nonn

%O 1,7

%A _Antti Karttunen_, Nov 08 2016