A036840 -id:A036840 - cp's OEIS frontend

A036845 a(n) = min_{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, 2, 2, 4, 2, 4, 4, 4, 4, 4, 4, 12, 4, 8, 8, 16, 4, 16, 8, 12, 4, 12, 8, 12, 12, 16, 12, 16, 8, 16, 16, 12, 16, 16, 12, 36, 16, 16, 16, 16, 12, 32, 12, 16, 12, 16, 16, 32, 12, 32, 16, 32, 16, 16, 16, 36, 16, 16, 16, 48, 16, 36, 32, 48, 12, 48, 32, 16, 16, 48, 16, 72, 36, 16

Offset: 1

Joseph L. Pe, Jan 09 2002

Conjecture: The sequence {T(n,k)} is eventually periodic for every n, so a(n) can be computed in finite time.

Conjecture: a(n) -> infinity as n -> infinity.

			The sequence {T(5,k)} is 4, 7, 6, 12, 4, 7, 6, 12,..., whose minimum value is 4. Hence a(5) = 4.

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

Cf. A000010, A000203, A036840, A066437, A096865.

a[ n_ ] := For[ m=EulerPhi[ n ]; min=Infinity; seq={m}, True, AppendTo[ seq, m ], If[ m

a(n) = A096865(A000010(n)). - Antti Karttunen, Dec 06 2017

Edited by Dean Hickerson, Jan 18 2002

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.

Original entry on oeis.org

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

Joseph L. Pe, Jan 08 2002

nonn

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

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

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

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

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 ] ] ]

(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

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

Edited by Dean Hickerson, Jan 18 2002

cp's OEIS Frontend

A036845 a(n) = min_{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.

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