A066437 -id:A066437 - cp's OEIS frontend

A096864 Function A062402(x) = sigma(phi(x)) is iterated. Starting with n, a(n) is the largest term arising in trajectory, either in transient or in terminal cycle.

1, 2, 3, 4, 12, 6, 12, 12, 12, 12, 18, 12, 28, 14, 15, 16, 72, 18, 72, 20, 28, 22, 36, 24, 42, 28, 72, 28, 72, 30, 72, 72, 42, 72, 72, 36, 252, 72, 72, 72, 90, 42, 252, 44, 72, 46, 72, 72, 252, 50, 252, 72, 252, 72, 90, 72, 252, 72, 90, 72, 168, 72, 252, 252, 168, 66, 168, 252

Offset: 1

Labos Elemer, Jul 21 2004

			n=256: list={256,255,255}, a(256)=256 as a transient term;
n=101: list={101,217,546,403,1170,819,[1240,1512],1240,...}, a(101)=1512 as a cycle term.

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

Cf. A062401, A062402, A066437, A096862, A096863, A096866 (smallest term), A096993.

Cf. also A096861.

gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Max[gite[w, 20]], {w, 1, 256}]
Table[Max[NestList[DivisorSigma[1,EulerPhi[#]]&,n,20]],{n,70}] (* Harvey P. Dale, May 13 2019 *)

(define (A096864 n) (let loop ((visited (list n)) (m n)) (let ((next (A062402 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (max m next))))))) ;; Antti Karttunen, Nov 18 2017

a(n) = max(n, A066437(n)). - Antti Karttunen, Dec 06 2017

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.

Original entry on oeis.org

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

A036840 a(n) is the average of the repeating terms of {T(n,k)} rounded to the nearest integer (rounding up if there's a choice), if {T(n,k)} is eventually periodic; = 0 otherwise. Here 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, 7, 3, 7, 7, 7, 7, 7, 7, 20, 7, 12, 12, 39, 7, 39, 12, 20, 7, 20, 12, 20, 20, 39, 20, 39, 12, 39, 39, 20, 39, 39, 20, 154, 39, 39, 39, 39, 20, 154, 20, 39, 20, 39, 39, 154, 20, 154, 39, 154, 39, 39, 39, 154, 39, 39, 39, 100, 39, 154, 154, 100, 20, 100, 154, 39, 39

Offset: 1

Joseph L. Pe, Jan 09 2002

nonn

Conjecture: a(n) is never 0; i.e. the sequence {T(n,k)} is eventually periodic for every n.

a(n) - n can be thought of as the final score in the phi/sigma tug-of-war with seed n. For example a(5) - 5 = 7 - 5 = 2, so sigma wins by "2 points" over phi at 5. a(8) - 8 = 7 - 8 = -1, so phi wins by "1 point" over sigma at 8. a(3) - 3 = 3 - 3 = 0, so it is a tie at 3. Are sigma's margins of victory over phi bounded? Are phi's bounded?

			The sequence {T(5,k)} is 4, 7, 6, 12, 4, 7, .... The average of the repeating numbers is 7.25 which rounds off to 7. So a(5) = 7. The sequence {T(37,k)} is 36, 91, 72, 195, 96, 252, 72, 195, .... The average of the repeating numbers is 153.75, which rounds off to 154. So a(37) = 154.

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

Cf. A000010, A000203, A036845, A066437, A095955, A096993.

a[ n_ ] := Module[ {}, For[ m=n; seq={}, !MemberQ[ seq, m ], m=DivisorSigma[ 1, EulerPhi[ m ] ], AppendTo[ seq, m ] ]; rp=Drop[ seq, Position[ seq, m ][ [ 1, 1 ] ]-1 ]; Floor[ 1/2+(Plus@@Join[ rp, EulerPhi/@rp ])/2/Length[ rp ] ] ]

(define (A036840 n) (let loop ((visited (list n)) (i 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)) (floor->exact (+ 1/2 (/ (apply + start_of_cyclic_part) (length start_of_cyclic_part))))) (else (loop (cons next visited) (+ 1 i)))))) (else (loop (cons next visited) (+ 1 i))))))) ;; Antti Karttunen, Dec 06 2017

Edited by Dean Hickerson, Jan 18 2002

cp's OEIS Frontend

A096864 Function A062402(x) = sigma(phi(x)) is iterated. Starting with n, a(n) is the largest term arising in trajectory, either in transient or in terminal cycle.

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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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