A277890 Number of even numbers encountered before (n^2)-1 is reached when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

0, 2, 0, 3, 2, 3, 1, 5, 3, 4, 4, 6, 3, 5, 3, 7, 8, 8, 6, 8, 9, 10, 6, 8, 10, 10, 7, 11, 10, 13, 11, 12, 12, 14, 10, 13, 12, 13, 14, 15, 13, 15, 15, 18, 18, 16, 15, 17, 21, 18, 18, 18, 19, 20, 16, 21, 20, 20, 22, 20, 23, 20, 22, 23, 21, 23, 23, 27, 25, 24, 22, 28, 22, 27, 24, 26, 25, 25, 29, 29, 28, 26, 30, 31, 28, 28, 31, 30, 32, 33, 27, 32, 34, 34, 30, 33, 33

Offset: 1

Author

Antti Karttunen, Nov 08 2016

Keywords

• nonn

Comments

The starting point ((n+1)^2)-1 of the iteration is included if it is even, but the ending point (n^2)-1 is never included in the count.

a(n) = number of even numbers on row n of A276574, after the initial zero-row.

See also comments in A277891.

Examples

			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 three of these numbers are even, thus a(6) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000035, A000290, A002828, A010052, A260734, A276573, A276574, A277486, A277488, A277889, A277891.

Programs

PARI

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
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From Charles R Greathouse IV, Jul 19 2011
A277890(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + (1-(k%2)); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277890.txt", n, " ", A277890(n)));

Scheme

(define (A277890 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 (- 1 (A000035 k))))))))

Formula

a(n) + A277891(n) = A260734(n).

For n >= 2, a(n) >= A277486(n).

a(n) >= A277488(n).