A062402 -id:A062402 - cp's OEIS frontend

A096861 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the largest term arising in trajectory.

1, 2, 3, 6, 5, 6, 7, 8, 12, 10, 11, 12, 13, 14, 15, 30, 17, 30, 19, 20, 30, 22, 23, 30, 30, 26, 30, 30, 29, 30, 31, 96, 33, 34, 35, 96, 37, 38, 39, 40, 41, 96, 43, 44, 45, 46, 47, 60, 96, 60, 51, 96, 53, 96, 55, 96, 96, 58, 59, 60, 61, 96, 63, 126, 65, 66, 96, 96, 96, 70, 71, 96

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=255: list={255,144,360,288,[432,480],432,...}, a(255)=480, a recurrent term;
n=247: list={247,96,72,96,...}, a(247)=247, a transient term, here the initial value.

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

Cf. A000010, A000203, A062401, A062402, A095955, A096859, A096865.

Cf. also A096864.

gf[x_] :=DivisorSigma[1, EulerPhi[x]] itef[x_, len_] :=NestList[fs, x, len] Table[Max[itef[w, 20]], {w, 1, 256}]

(define (A096861 n) (let loop ((visited (list n)) (m n)) (let ((next (A062401 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (max m next))))))) ;; Antti Karttunen, Nov 18 2017

A096994 Number of transient terms if f(x)=phi(sigma(x))=A062401 is iterated at initial value 2^n. Equilibrium terms are listed in A096852.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 4, 1, 2, 5, 14, 0, 5, 7, 2, 14, 8, 3, 64, 43, 81, 82, 76, 74, 47, 25, 42, 0

Offset: 0

Labos Elemer, Jul 22 2004

For transient lengths for iterations of A062401(x) or A062402(x) if started at 2^n, A096994(n) + 1 = A096995(n). Corresponding cycle lengths satisfy A096852(n-1) = A096857(n). Behind these observations several relationships stand, e.g., sigma(A062401(x)) = A062402(sigma(x)) or phi(A062402(x)) = A062401(phi(x)).

			n=0: trajectory = {1,1,..} so a(0)=0;
n=14: transient-length=14, cycle-length=2, a(14)=14, A096852(14)=2; trajectory ={16384, 27000, 23040, 21600, 17280, 15360, 15488, 13824, 9600, 7680, 7200, 12960, 11880, 11520, [10368,14080], 10368, ...}.
Values of a(n) for n > 31, with -1 signifying transient lengths yet unknown after 10^4 iterations of f(x): -1, 7, 51, 70, 23, 39, 11, -1, 37, 107, 30, -1, 145, 25, 21, 36, -1, -1, -1, -1, 31, -1, 452, -1, 449, 447, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 40, -1, -1, -1, -1, -1, -1, -1, 279, -1. - _Michael De Vlieger_, May 15 2017

Cf. A096852, A062401.

With[{nn = 10^3}, Table[Count[Values@ PositionIndex@ #, k_ /; Length@ k == 1] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, 2^n, nn] /. k_ /; k == nn + 1 -> -1, {n, 31}] ] (* Michael De Vlieger, May 15 2017, Version 10 *)

A230201 Numbers k such that sigma(phi(k)) < k.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 46, 48, 50, 54, 58, 60, 64, 66, 68, 70, 72, 78, 80, 84, 90, 92, 94, 96, 98, 100, 102, 106, 108, 110, 114, 118, 120, 126, 128, 130, 132, 136, 138, 140, 144, 150, 156, 160, 162, 166, 168

Offset: 1

Vladimir Letsko, Oct 11 2013

nonn

All terms are even. However, sigma(phi(k)) may be equal to k for an odd number k if k+2 is a Fermat prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001229, A018784, A062402, A066694, A227011, A227927, A230203.

Maple
```
for n do if sigma(phi(n))
				
```

Select[Range[200], DivisorSigma[1, EulerPhi[#]] < # &] (* T. D. Noe, Oct 14 2013 *)

isok(n) = sigma(eulerphi(n)) < n; \\ Michel Marcus, Oct 12 2013

A065391 Numbers m such that A062401(m) = phi(sigma(m)) is increasing to a record value, i.e., A062401(m) represents a new peak, so that A062401(m) > A062401(k) for all k < m.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 32, 36, 64, 100, 144, 256, 324, 400, 576, 900, 1296, 1600, 2304, 2916, 3600, 5184, 8100, 9216, 11664, 14400, 20736, 22500, 25600, 30276, 32400, 41616, 46656, 57600, 69696, 72900, 82944, 90000, 104976, 115600, 121104, 129600

Offset: 1

Labos Elemer, Nov 05 2001

nonn

The terms > 2 are exact powers and except for 2, 8 and 32 all the terms seem to be squares.

Indices of records in A062401. - Michael De Vlieger, Dec 06 2018

			Initial segment of A062401: {1, 2, 2, 6, 2, 4, 4, 8, 12, 6, 4, 12, 6, 8, 8, 30, 6, ...}. The peak values (those exceeding all previous ones) are 1, 2, 6, 8, 12, 30, reached at positions 1, 2, 4, 8, 9, 16, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..249 (terms below 10^11; terms 1..100 from Harry J. Smith)

Cf. A000010, A000203, A062401, A062402, A065392.

a = 0; s = 0; Do[s = EulerPhi[DivisorSigma[1, n]]; If[s > a, a = s; Print[n]], {n, 1, 10^6}]
(* Second program: *)
With[{s = Array[EulerPhi@ DivisorSigma[1, #] &, 2*10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Dec 06 2018 *)
DeleteDuplicates[Table[{n,EulerPhi[DivisorSigma[1,n]]},{n,150000}],GreaterEqual[ #1[[2]],#2[[2]]]&] [[;;,1]] (* Harvey P. Dale, May 12 2023 *)

{ n=r=0; for (m=1, 10^9, x=eulerphi(sigma(m)); if (x > r, r=x; write("b065391.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009

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

A066930 Numbers k such that phi(sigma(k)) divides sigma(phi(k)).

Original entry on oeis.org

1, 7, 9, 29, 71, 97, 109, 121, 139, 142, 175, 183, 194, 215, 225, 242, 244, 261, 278, 311, 344, 349, 355, 430, 497, 509, 516, 533, 556, 571, 605, 622, 631, 647, 673, 709, 729, 791, 817, 859, 911, 923, 1021, 1066, 1112, 1119, 1142, 1207, 1243, 1262, 1277

Offset: 1

Benoit Cloitre, Jan 26 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010 (phi), A000203 (sigma), A033632 (subsequence), A062401, A062402.

Select[Range[1300],Divisible[DivisorSigma[1,EulerPhi[#]],EulerPhi[ DivisorSigma[ 1,#]]]&] (* Harvey P. Dale, Dec 15 2021 *)

isok(k) = { sigma(eulerphi(k)) % eulerphi(sigma(k)) == 0 } \\ Harry J. Smith, Apr 08 2010

A096853 a(n) = A062401(2^n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 48, 64, 144, 288, 512, 576, 2304, 4096, 10240, 18432, 36288, 65536, 184320, 262144, 552960, 718848, 1492992, 2822400, 9123840, 13418496, 44695552, 68762880, 106168320, 109486080, 580386816, 1073741824, 2155507200, 2366668800, 6920601600, 12081954816

Offset: 1

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..392

Cf. A000010, A000203, A000225, A062401, A062402, A053287, A075708, A096853, A096854, A096855, A096856.

Table[ EulerPhi[ DivisorSigma[1, 2^n - 1]], {n, 33}]

a(n) = A000010(A000203(A000225(n))). - Michel Marcus, Dec 19 2013

a(n) = A000010(A075708(n)). - Amiram Eldar, Jun 04 2024

Edited and extended by Robert G. Wilson v, Jul 23 2004

a(33)-a(35) from Amiram Eldar, Jun 04 2024

A096855 a(n) = A062401(2^n + 1).

Original entry on oeis.org

2, 2, 2, 12, 6, 16, 24, 80, 84, 320, 360, 864, 1320, 5456, 5184, 15744, 19800, 52800, 69120, 349520, 370080, 1036800, 1425600, 3640896, 4741632, 13989888, 27091584, 76743040, 94656000, 166387200, 412473600, 1407389952, 1420488192, 3459760128, 6502788864, 14778408960

Offset: 0

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..400

Cf. A000010, A062401, A062402, A053287, A096853, A069061, A096854, A096855, A096856.

Table[ EulerPhi[ DivisorSigma[1, 2^n+1]], {n, 0, 33}]

a(n) = A000010(A069061(n)). - Amiram Eldar, Jun 04 2024

Edited and extended by Robert G. Wilson v, Jul 23 2004

Offset changed to 0, a(0) prepended and two more terms added by Amiram Eldar, Jun 04 2024

A096860 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in the transient of this trajectory, that is: a(n) = A096859(n) - A095955(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 0, 2, 1, 1, 4, 2, 3, 1, 3, 3, 1, 1, 0, 1, 3, 1, 2, 1, 1, 3, 2, 3, 1, 1, 0, 2, 2, 1, 1, 3, 1, 3, 1, 2, 2, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 3, 1, 3, 0, 4, 2, 1, 5, 3, 1, 1, 1, 3

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=255: list={255,144,360,288,[432,480],432,...}, t=transient=4, c=cycle=2, a(255)=t=4;
n=244: list={244,180,144,360,288,[432,480],432,...}, a(244)=4.
a(n)=0 means that n is a recurrent term from A096850.

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

Cf. A000010, A000203, A062401, A062402, A095955, A096859, A096861, A096865.

Cf. also A096863.

With[{nn = 120}, Array[Length@ Union@ # - Length@ Select[Tally@ #, Last@ # > 1 &] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, #, nn] &, 105]] (* Michael De Vlieger, Nov 18 2017 *)

(define (A096860 n) (let loop ((visited (list n))) (let ((next (A062401 (car visited)))) (cond ((member next visited) => (lambda (transientplusone) (- (length transientplusone) 1))) (else (loop (cons next visited))))))) ;; Antti Karttunen, Nov 18 2017

A227011 Integers m such that phi(sigma(k))/sigma(phi(k)) > phi(sigma(m))/sigma(phi(m)) for all k

Original entry on oeis.org

1, 3, 5, 11, 13, 17, 29, 41, 181, 209, 377, 779, 3239, 4469, 5249, 15539, 43259, 58589, 119279, 169679, 174719, 461369, 692687, 955499, 1258949, 1859129, 1917299, 3925463, 7991693, 8986469, 13244069, 16732169, 30629363, 44137523, 48466987, 64018433, 68787773

Offset: 1

Vladimir Letsko, Oct 09 2013

nonn

These are the indices where the rational function A062401(n)/A062402(n) drops below the minimum set by all earlier ratios.

a(2) to a(9) are primes. However all known terms beginning from a(10) are composite.

			5 is in the sequence because phi(sigma(5))/sigma(phi(5)) = 2/7 and for all k < 5, phi(sigma(k))/sigma(phi(k)) > 2/7.

Donovan Johnson, Table of n, a(n) for n = 1..48 (terms < 10^10)

Cf. A227927, A062401, A062402, A033632, A229238.

A062401 := proc(n)
    numtheory[phi](numtheory[sigma](n))
end proc:
A062402 := proc(n)
    numtheory[sigma](numtheory[phi](n))
end proc:
s := proc(n)
    A062401(n)/A062402(n) ;
end proc:
r := 100000000000000000000000000000 ;
for n from 1 do
    if s(n) < r then
        printf("%d,\n",n) ;
        r := s(n) ;
    end if;
end do:

f(n)=eulerphi(sigma(n=factor(n)))/sigma(eulerphi(n))
is(n)=my(t=f(n)); for(k=1,n-1,if(f(k)<=t, return(0))); 1 \\ Charles R Greathouse IV, Nov 27 2013

a(33)-a(37) from Donovan Johnson, Oct 11 2013

cp's OEIS Frontend

A096861 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the largest term arising in trajectory.

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A096994 Number of transient terms if f(x)=phi(sigma(x))=A062401 is iterated at initial value 2^n. Equilibrium terms are listed in A096852.

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A230201 Numbers k such that sigma(phi(k)) < k.

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A065391 Numbers m such that A062401(m) = phi(sigma(m)) is increasing to a record value, i.e., A062401(m) represents a new peak, so that A062401(m) > A062401(k) for all k < m.

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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.

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A066930 Numbers k such that phi(sigma(k)) divides sigma(phi(k)).

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A096853 a(n) = A062401(2^n-1).

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A096855 a(n) = A062401(2^n + 1).

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