A077080 -id:A077080 - cp's OEIS frontend

A077085 Initial values such that if A077080(x)=phi(sigma(x)+phi(x)) is started at these numbers then the sequence does not converge.

534, 556, 557, 580, 624, 702, 710, 738, 739, 740, 748, 784, 789, 822, 823, 841, 852, 853, 900, 912, 913, 916, 924, 931, 938, 960, 961, 962, 1020, 1021, 1029, 1032, 1033, 1034, 1065, 1089, 1092, 1093, 1098, 1126, 1136

Offset: 1

Labos Elemer, Oct 28 2002

nonn

These terms are only conjectures.

These terms survive 1000 iterations. - Sean A. Irvine, May 05 2025

Cf. A000010, A000203, A065387, A077080, A077081, A077082, A077083, A077084.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Do[s=Part[NestList[f, n, 100], 100]; If[Greater[s, 10000000], Print[{n, s}]], {n, 1, 10000}]

A077081 Fixed point when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

Original entry on oeis.org

1, 2, 2, 6, 6, 6, 6, 864, 864, 10, 10, 864, 864, 864, 864, 864, 864, 864, 864, 20, 20, 22, 22, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 48, 864, 46, 46, 48, 864, 864, 48, 864, 864, 48, 48, 48, 48, 58, 58

Offset: 1

Labos Elemer, Oct 28 2002

nonn

A065387 when iterated seems to converge [tested for initial values below 1024]. On the other hand iterating A051682 often ends in cycle.
Iteration of phi(A065387())=phi(sigma()+phi()) seems to converge. Tested below n=1024. Critical values however arise. For example: n=534,556,557,580,624,702,710, etc. These initial values generate very large terms and i was unable to decide if they converge.
For n=1..1024 no more but 27 distinct fixed points arised:{1,2,6,10,..,3552,570240}

			n=225: results in iteration sequence of 44 terms: {225,522,444,...,471744,653312,570240}, a[25]=570240.

Cf. A000010, A000203, A065387, A077080, A077082, A077083, A077084, A077085.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Table[FixedPoint[f, w], {w, 1, 256}]

a(n) = FixedPoint[A077080, n].

A077082 Largest value arising when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

Original entry on oeis.org

1, 2, 3, 6, 6, 6, 7, 1044, 1044, 10, 11, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 20, 21, 22, 23, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 48, 1044, 46, 47, 48, 1044

Offset: 1

Labos Elemer, Oct 28 2002

nonn

			n=225: results in iteration sequence of 44 terms: {225,522,444,...,471744,653312,570240}, largest is 653312=a(225).

Cf. A000010, A000203, A065387, A077080-A077085.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Table[Max[FixedPointList[f, w]], {w, 1, 1024}]

a(n) = Max[FixedPointList[A077080, n]]. See program below. Seems convergent. [tested for initial values below 1024.]

A077083 Length of iteration until a fixed point is reached when phi(sigma[n]+phi(n)) = A077080(n) is iterated with initial value of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 2, 16, 16, 1, 2, 16, 17, 17, 16, 15, 16, 15, 16, 1, 2, 1, 2, 14, 14, 16, 15, 14, 15, 14, 15, 13, 14, 15, 15, 10, 11, 15, 14, 13, 14, 11, 12, 2, 14, 1, 2, 1, 12, 6, 2, 12, 13, 3, 2, 2, 3, 1, 2, 11, 12, 11, 2, 11, 14, 10, 11, 13, 2, 2, 3, 5, 6, 14, 10, 10, 2, 12, 13, 9

Offset: 1

Labos Elemer, Oct 28 2002

nonn

			n=1,2,6,10,..864: a[n]=1, n is fixed point for some n; n=225: results in iteration sequence of 44 terms: {225,522,444,...,471744,653312,570240}, 44=a[225].

Cf. A000010, A000203, A065387, A077080-A077085.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Table[Length[FixedPointList[f, w]]-1, {w, 1, 1024}]

a(n)=Length[FixedPointList[A077080, n]]-1.

A097646 Numbers n such that n = phi(phi(n) + sigma(n)).

Original entry on oeis.org

1, 2, 6, 10, 20, 22, 46, 48, 58, 82, 106, 166, 178, 180, 208, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 864, 886, 982, 1018, 1120, 1186, 1282, 1306, 1318, 1366, 1368, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206

Offset: 1

Farideh Firoozbakht, Sep 08 2004

nonn

If n=2*p where p is a Sophie Germain odd prime, then n is in the sequence; the proof is obvious.

			22 is in the sequence because phi(22)=10, sigma(22)=36 and phi(10+36)=22.

K. D. Bajpai, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Glossary, Sophie Germain prime.

Cf. A077080, A097652, A005384, A097645, A018784.

[n: n in [1..2300] | n eq EulerPhi(EulerPhi(n) + DivisorSigma(1,n))]; // Vincenzo Librandi, Aug 22 2015

with(numtheory):K:=proc()local n,a,c;  c:=1; for n from 1 to 5000000 do;
a:=phi(phi(n)+ sigma(n));if  a=n  then lprint(c,n); c:=c+1; fi;od; end:K(); # K. D. Bajpai, Jul 18 2013

Do[If[n==EulerPhi[EulerPhi[n]+DivisorSigma[1, n]], Print[n]], {n, 2400}]
Select[Range[2500],EulerPhi[EulerPhi[#]+DivisorSigma[1,#]]==#&] (* Harvey P. Dale, Jul 06 2021 *)

is(n)=sigma(n=factor(n))==eulerphi(eulerphi(n)) \\ Charles R Greathouse IV, Nov 27 2013

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A077085 Initial values such that if A077080(x)=phi(sigma(x)+phi(x)) is started at these numbers then the sequence does not converge.

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A077081 Fixed point when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

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A077082 Largest value arising when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

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A077083 Length of iteration until a fixed point is reached when phi(sigma[n]+phi(n)) = A077080(n) is iterated with initial value of n.

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A097646 Numbers n such that n = phi(phi(n) + sigma(n)).

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