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A278515 Number of steps required to iterate map k -> A255131(k) when starting from k = (A000196(n)+1)^2 before n is reached, or 0 if n is not reached.

%I A278515 #14 Jul 20 2025 15:03:44
%S A278515 1,0,0,1,0,0,2,0,1,0,0,2,0,0,0,1,4,0,3,0,0,2,0,0,1,0,0,4,0,0,3,0,2,0,
%T A278515 0,1,0,0,5,0,4,0,0,3,0,2,0,0,1,0,0,5,0,4,0,0,3,0,0,2,0,0,0,1,7,0,0,6,
%U A278515 0,0,5,0,4,0,0,3,0,0,2,0,1,0,0,7,0,6,0,0,5,0,4,0,0,3,0,0,2,0,0,1,0,0,7,0,0,6,0,0,5,0,0,0,4,0,0,3,0,2,0,0,1
%N A278515 Number of steps required to iterate map k -> A255131(k) when starting from k = (A000196(n)+1)^2 before n is reached, or 0 if n is not reached.
%H A278515 Antti Karttunen, <a href="/A278515/b278515.txt">Table of n, a(n) for n = 0..10200</a>
%e A278515 For n=15, we start iterating from the next larger square, (floor(sqrt(15))+1)^2 = 16, and in just a single step (16 - A002828(16) = 15) we land to n, so a(15) = 1.
%e A278515 For n=16, we start iterating from the next larger square, which is 25, and thus we have 25 -> A255131(25) = 24, 24 -> A255131(24) = 21, 21 -> A255131(21) = 18, 18 -> A255131(18) = 16, thus four steps were required to reach 16, and a(16) = 4.
%e A278515 For n=17, when we do the same iteration, we see that we will pass by it, thus a(17) = 0.
%o A278515 (Scheme) (define (A278515 n) (let* ((r (A000196 n)) (end (A000290 r))) (let loop ((k (A000290 (+ 1 r))) (s 0)) (cond ((= k n) s) ((<= k end) 0) (else (loop (A255131 k) (+ 1 s)))))))
%Y A278515 Cf. A000196, A000290, A002828, A255131.
%Y A278515 Cf. A005563 (positions of ones), A276573 (of nonzero terms).
%K A278515 nonn
%O A278515 0,7
%A A278515 _Antti Karttunen_, Nov 28 2016