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%I #21 Apr 15 2019 01:55:55
%S 1,2,3,4,5,6,16,7,17,8,15,32,12,10,18,20,19,256,9,14,34,48,40,50,33,
%T 60,257,11,97,258,13,101,209,65536,21,259,64,30,65,51,80,24,84,36,85,
%U 66,260,22,4352,26,4368,28,4369,37,768,41,770,42,771,68,90,272,45,273,56,1200,952,4096,23,4097,27,4098,86,512,54
%N Start with a(1) = 1, then always choose for a(n) the least unused number such that A193231(a(n)*a(n-1)) = A193231(a(n)) * A193231(a(n-1)), where A193231 is an involution of natural numbers called Blue code.
%C Does this sequence die after a(144) = 46 ?
%C No, a(145) = 16777216, but whether the sequence is finished remains open. - _Rémy Sigrist_, Feb 15 2019
%C The next unused number of the form 2^2^k is always a valid choice, so this sequence is infinite. - _Charlie Neder_, Apr 14 2019
%H Rémy Sigrist, <a href="/A265405/b265405.txt">Table of n, a(n) for n = 1..183</a> (first 144 terms from Antti Karttunen) [More terms needed for b-file.]
%H Rémy Sigrist, <a href="/A265405/a265405.gp.txt">PARI program for A265405</a>
%o (Scheme, with defineperm1-macro from Antti Karttunen's IntSeq-library)
%o (defineperm1 (A265405 n) (cond ((= 1 n) n) (else (let ((prev (A265405 (- n 1)))) (let loop ((k 1)) (cond ((and (not-lte? (A265406 k) (- n 1)) (= (A193231 (* k prev)) (* (A193231 k) (A193231 prev)))) k) (else (loop (+ 1 k)))))))))
%o ;; We consider a > b (i.e. not less than b) also in case a is #f.
%o ;; (Because of the stateful caching system used by defineperm1-macro):
%o (define (not-lte? a b) (cond ((not (number? a)) #t) (else (> a b))))
%o (PARI) See Links section.
%Y Inverse: A265406.
%Y Cf. A193231.
%Y Cf. A266195, A266351, A266405 (sequences with similar definitions, of which at least the first two are known to be infinite and also bijective).
%K nonn,base
%O 1,2
%A _Antti Karttunen_, Dec 29 2015
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