A066437 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.

1, 1, 3, 3, 12, 3, 12, 12, 12, 12, 18, 12, 28, 12, 15, 15, 72, 12, 72, 15, 28, 18, 36, 15, 42, 28, 72, 28, 72, 15, 72, 72, 42, 72, 72, 28, 252, 72, 72, 72, 90, 28, 252, 42, 72, 36, 72, 72, 252, 42, 252, 72, 252, 72, 90, 72, 252, 72, 90, 72, 168, 72, 252, 252, 168, 42

Offset

1,3

Comments

Conjecture: a(n) is always finite; i.e. the sequence {T(n,k)} is eventually periodic for every n.

a(n) >= sigma(phi(n)) >= phi(n); since phi(n) -> infinity with n, so does a(n).

Sequence is otherwise like A096864, except here the initial value n where the iteration is started from is ignored. - Antti Karttunen, Dec 06 2017

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = A096864(A062402(n)). - Antti Karttunen, Dec 06 2017

Example

For n=11, the sequence is 11, 10, 18, 6, 12, 4, 7, 6, 12, ..., whose maximum value is 18. Hence a(11) = 18.

Mathematica

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 ] ] ]

Prog

(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

Crossrefs

Cf. A000010, A000203, A036840, A036845, A062402, A096864.

Sequence in context: A211346 A163372 A349922 * A153270 A065957 A303309

Adjacent sequences: A066434 A066435 A066436 * A066438 A066439 A066440

Keyword

nonn

Author

Joseph L. Pe, Jan 08 2002

Extensions

Edited by Dean Hickerson, Jan 18 2002

Status

approved