A277486 - OEIS

%I #10 Nov 14 2016 10:35:19

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

%T 3,4,3,4,5,5,5,4,4,6,6,3,3,9,4,5,6,9,4,6,4,4,8,6,5,7,5,9,5,5,7,8,6,11,

%U 5,9,4,7,9,9,6,10,5,5,17,4,10,9,10,7,3,3,10,8,7,10,6,9,5,10,10,10,8,11,6,9,10,7,7,7,7,12,9,11,13,9,12,6,10,9,6

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

%H Antti Karttunen, <a href="/A277486/b277486.txt">Table of n, a(n) for n = 1..10000</a>

%F For n >= 2, a(n) <= A277890(n).

%e For n=6, we start iterating from k = ((6+1)^2)-1 = 48, and then 48 - A002828(48) = 45, 45 - A002828(45) = 43, 43 - A002828(43) = 40, 40 - A002828(40) = 38, and 38 - A002828(38) = 35 (which is 6^2 - 1), and when we subtract one from each, only 47 and 37 are primes, thus a(6) = 2.

%e For n=7, we start iterating from k = ((7+1)^2)-1 = 63, and 63 -> 59, 59 -> 56, 56 -> 53, 53 -> 51, 51 -> 48 (which is 7^2 - 1), and subtracting one from each of 63, 59, 56, 53 and 51, doesn't yield a prime for any, thus a(7)=0. (Note that even though 48-1 = 47 is a prime, it is not included in the count for n=7).

%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 A277486(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + if(isprime(k-1),1,0); k = k - A002828(k)); s; };

%o for(n=1, 10000, write("b277486.txt", n, " ", A277486(n)));

%o (Scheme)

%o (define (A277486 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 (+ -1 k))))))))

%Y Cf. A000290, A010051, A010052, A277487, A277488, A277890, A278166 (partial sums).

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 08 2016