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%I #15 Dec 06 2017 20:08:01
%S 1,1,3,3,12,3,12,12,12,12,18,12,28,12,15,15,72,12,72,15,28,18,36,15,
%T 42,28,72,28,72,15,72,72,42,72,72,28,252,72,72,72,90,28,252,42,72,36,
%U 72,72,252,42,252,72,252,72,90,72,252,72,90,72,168,72,252,252,168,42
%N a(n) = max_{k} {T(n,k)} where T(n,k) is the "phi/sigma tug-of-war sequence with seed n" defined by T(n,1) = phi(n), T(n,2) = sigma(phi(n)), T(n,3) = phi(sigma(phi(n))), ..., T(n,k) = phi(T(n,k-1)) if k is odd and = sigma(T(n,k-1)) if k is even.
%C Conjecture: a(n) is always finite; i.e. the sequence {T(n,k)} is eventually periodic for every n.
%C a(n) >= sigma(phi(n)) >= phi(n); since phi(n) -> infinity with n, so does a(n).
%C Sequence is otherwise like A096864, except here the initial value n where the iteration is started from is ignored. - _Antti Karttunen_, Dec 06 2017
%H Antti Karttunen, <a href="/A066437/b066437.txt">Table of n, a(n) for n = 1..16384</a>
%F a(n) = A096864(A062402(n)). - _Antti Karttunen_, Dec 06 2017
%e For n=11, the sequence is 11, 10, 18, 6, 12, 4, 7, 6, 12, ..., whose maximum value is 18. Hence a(11) = 18.
%t a[ n_ ] := For[ m=n; max=0; seq={}, True, AppendTo[ seq, m ], If[ (m=DivisorSigma[ 1, EulerPhi[ m ] ])>max, max=m ]; If[ MemberQ[ seq, m ], Return[ max ] ] ]
%o (Scheme) (define (A066437 n) (let loop ((visited (list n)) (i 1) (m 1)) (let ((next ((if (odd? i) A000010 A000203) (car visited)))) (cond ((member next (reverse visited)) => (lambda (start_of_cyclic_part) (cond ((even? (length start_of_cyclic_part)) (max m next)) (else (loop (cons next visited) (+ 1 i) (max m next)))))) (else (loop (cons next visited) (+ 1 i) (max m next))))))) ;; _Antti Karttunen_, Dec 06 2017
%Y Cf. A000010, A000203, A036840, A036845, A062402, A096864.
%K nonn
%O 1,3
%A _Joseph L. Pe_, Jan 08 2002
%E Edited by _Dean Hickerson_, Jan 18 2002
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