A171262 -id:A171262 - cp's OEIS frontend

A171271 Numbers n such that phi(n)=2*phi(n-1).

3, 5, 17, 155, 257, 287, 365, 805, 1067, 2147, 3383, 4551, 6107, 7701, 8177, 9269, 11285, 12557, 12971, 16403, 19229, 19277, 20273, 25133, 26405, 27347, 29155, 29575, 35645, 36419, 38369, 39647, 40495, 47215, 52235, 54653, 65537, 84863

Offset: 1

Farideh Firoozbakht, Feb 23 2010

Theorem: A prime p is in the sequence iff p is a Fermat prime.

Proof: If p=2^2^n+1 is prime (Fermat prime) then phi(p)=2^2^n=2* phi(2^2^n)=2*phi(p-1), so p is in the sequence. Now if p is a prime term of the sequence then phi(p)=2*phi(p-1) so p-1=2*phi(p-1) and we deduce that p-1=2^m hence p is a Fermat prime.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 5416 terms from Hiroaki Yamanouchi)

Cf. A019434, A050472, A171262.

[n: n in [2..2*10^5] | EulerPhi(n) eq 2*EulerPhi(n-1)]; // Vincenzo Librandi, May 17 2015

Select[Range[85000],EulerPhi[ # ]==2EulerPhi[ #-1]&]
Flatten[Position[Partition[EulerPhi[Range[90000]],2,1],?(2#[[1]] == #[[2]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Sep 09 2017 *)

a(n) = A050472(n) + 1. - Ray Chandler, May 01 2015

A256907 Numbers n such that phi(n) = 3*phi(n+1).

Original entry on oeis.org

119, 527, 545, 2849, 3689, 4487, 6649, 18619, 26771, 30377, 44659, 47585, 50507, 76997, 83021, 102167, 112463, 128933, 138773, 163877, 174437, 192881, 193115, 198263, 217967, 236441, 243827, 244001, 254539, 268067, 282359, 287825, 298115, 345059, 410123, 464645

Offset: 1

Mauro Fiorentini, Apr 12 2015

nonn

			phi(545) = 3*phi(546) = 432.

Mauro Fiorentini, Table of n, a(n) for n = 1..755 (all terms with n <= 10^9).

Cf. A171262.

[n: n in [1..2*10^6] | EulerPhi(n) eq 3*EulerPhi(n+1)]; // Vincenzo Librandi, Apr 13 2015

Select[Range@ 1000000, EulerPhi@# == 3 EulerPhi[# + 1] &] (* Michael De Vlieger, Apr 12 2015 *)

s=[]; for(n=1, 1000000, if(eulerphi(n)==3*eulerphi(n+1), s=concat(s, n))); s \\ Colin Barker, Apr 12 2015

A256937 Numbers n such that phi(n) = 4*phi(n+1).

Original entry on oeis.org

629, 1469, 85139, 100889, 139859, 154979, 168149, 304079, 396899, 838199, 1107413, 1323449, 1465463, 2088839, 2160899, 2504879, 2684879, 2693249, 2800181, 3404609, 3512249, 3576869, 3885881, 4241819, 4500509, 4620659, 4822649, 5530709, 5805449

Offset: 1

Mauro Fiorentini, Apr 13 2015

nonn

			phi(629) = 576 = 4*phi(630).

Mauro Fiorentini, Table of n, a(n) for n = 1..154 (all terms for n up to 10^9).

Cf. A171262, A256907.

[n: n in [1..10^7] | EulerPhi(n) eq 4*EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015

A:= NULL:
y:= numtheory:-phi(1):
for n from 1 to 10^6 do
x:= numtheory:-phi(n+1);
if y = 4*x then A:= A, n fi;
y:= x;
od:
A;  # Robert Israel, Apr 15 2015

Select[Range@ 1000000, EulerPhi@ # == 4 EulerPhi[# + 1] &] (* Michael De Vlieger, Apr 13 2015 *)
Position[Partition[EulerPhi[Range[6*10^6]],2,1],?(#[[1]]==4#[[2]]&),{1},Heads->False]//Flatten (* _Harvey P. Dale, Sep 18 2016 *)

s=[]; for(n=1, 1000000, if(eulerphi(n)==4*eulerphi(n+1), s=concat(s, n))); s \\ Colin Barker, Apr 13 2015

[n for n in (1..1000000) if euler_phi(n) == 4*euler_phi(n+1)]; # Bruno Berselli, Apr 14 2015

A257550 Numbers n such that phi(n) = 5*phi(n+1).

Original entry on oeis.org

17907119, 18828809, 31692569, 73421039, 179467469, 322757819, 337567229, 627702389, 975314339, 2537636009, 2722271369, 3328653509, 3917646809, 5529412349, 6369847469, 11179199849, 11201693579, 11363832479, 13442120999, 16781760449, 19751331599, 20002320029

Offset: 1

Ray Chandler, Apr 29 2015

nonn

			phi(17907119) = 16588800 = 5*phi(17907120).

Giovanni Resta, Table of n, a(n) for n = 1..103 (terms < 10^12)

Cf. A001274, A171262, A256907, A256937.

a1={};nmax=10^9;last=EulerPhi[1];n=2;
While[nRay Chandler, Apr 30 2015 *)

a(10)-a(22) from Giovanni Resta, May 11 2015

A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

Original entry on oeis.org

1, 5, 119, 629, 17907119

Offset: 1

Felix Fröhlich, May 11 2015

From Manfred Scheucher, May 27 2015: (Start)
a(6)>=3*10^8 (calculation)
a(7)>=3.5*10^13, a(8)>=4.5*10^25, a(9)>=3.0*10^47, and so on... (doubly exponential lower bound, see uploaded pdf)
239719159679 and 239742643139 admit a ratio of 5.998... and 6.008..., resp.
There might be a relation to the sequence A098026. (End)

			a(3) = 119, because phi(119) == 3*phi(120) = 96 and 119 is the smallest k where this equality holds for n = 3.

Manfred Scheucher, A lower bound on A257865(n)

Cf. A001274, A171262, A256907, A256937, A257550.

Table[k = 1; While[EulerPhi[k] != n EulerPhi[k + 1], k++]; k, {n, 4}] (* Michael De Vlieger, May 12 2015 *)

a(n) = my(k=1); while(eulerphi(k)!=n*eulerphi(k+1), k++); k

a(n) >= exp(exp(c(n-3))) with c=exp(gamma) and gamma being the Euler-Mascheroni_constant (see pdf). - Manfred Scheucher, May 27 2015

A258454 Numbers n such that phi(n) = 2*phi(n-2).

Original entry on oeis.org

3, 4, 5, 8, 17, 32, 257, 512, 527, 992, 1952, 2522, 5252, 6512, 7412, 10376, 23432, 23717, 26732, 27302, 35114, 36632, 37442, 45872, 47027, 49022, 51092, 65537, 78899, 84242, 92432, 98432, 98672, 114767, 115292, 131072, 227222, 231167, 240977, 328352, 369272

Offset: 1

Jaroslav Krizek, Sep 24 2015

nonn

Conjecture: a prime p is in the sequence iff p is a Fermat prime (A019434).

This is not correct: the first non-Fermat prime term is 83623937 = 2^18*11*29 + 1. - Joerg Arndt, Oct 11 2015

			phi(32) = 16 = 2*phi(30) = 2*8, so 32 is in the sequence.

Cf. A171271, A171262.

[n: n in [3..10000000] | EulerPhi(n) eq 2*EulerPhi(n-2)];

Select[Range@ 400000, EulerPhi@ # == 2 EulerPhi[# - 2] &] (* Michael De Vlieger, Sep 25 2015 *)

for(n=1, 1e6, if(eulerphi(n) == 2*eulerphi(n-2), print1(n", "))); \\ Altug Alkan, Sep 26 2015

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A171271 Numbers n such that phi(n)=2*phi(n-1).

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A256907 Numbers n such that phi(n) = 3*phi(n+1).

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A256937 Numbers n such that phi(n) = 4*phi(n+1).

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A257550 Numbers n such that phi(n) = 5*phi(n+1).

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A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

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A258454 Numbers n such that phi(n) = 2*phi(n-2).

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