A166955 -id:A166955 - cp's OEIS frontend

A065528 Numbers n such that phi(n) is a nontrivial power b^c where b > 1 and c > 1.

5, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 37, 40, 48, 51, 57, 60, 63, 64, 68, 74, 76, 80, 85, 96, 101, 102, 108, 114, 120, 125, 126, 128, 136, 160, 170, 185, 192, 197, 202, 204, 219, 240, 247, 250, 255, 256, 257, 259, 272, 273, 285, 292, 296, 304, 315, 320, 327, 333

Offset: 1

Joseph L. Pe, Nov 27 2001

What values of b can occur?

Apparently all even numbers can occur as values of b. Checked up to 50000; see A227533. - Charles R Greathouse IV, Jul 15 2013

			phi(63) = 6^2, phi(96) = 2^5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A166955.

ppQ[n_] := GCD @@ Last /@ FactorInteger@ n > 1; Select[ Range@ 330, ppQ@ EulerPhi@ # &] (* Robert G. Wilson v, Jul 16 2013 *)

v=[]; for(n=2, 333, if(ispower(eulerphi(n)), v=concat(v, n))); v (Hobson)

is(n)=ispower(eulerphi(n)) \\ Charles R Greathouse IV, Jul 15 2013

a(n) = A166955(n+2). - Juri-Stepan Gerasimov, Oct 25 2009

More terms from Nick Hobson, Nov 29 2006

b-file from Charles R Greathouse IV, Mar 25 2010

A216412 The cubes arising in A039771.

Original entry on oeis.org

1, 1, 8, 8, 8, 8, 8, 64, 64, 64, 64, 64, 64, 64, 64, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 216, 512, 216, 216, 512, 512, 512, 1000, 1000, 512, 512, 1000, 512, 512, 512, 1728, 1728, 1000, 512, 1000, 512, 1728, 1000, 1728, 1728, 1000, 1000, 1728, 1728, 1000, 1728, 1728, 1000, 1728

Offset: 1

V. Raman, Sep 07 2012

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A039770, A039771, A039772, A062732, A078164, A166955.

Select[EulerPhi @ Range[3000], IntegerQ[Surd[#, 3]] &] (* Amiram Eldar, Mar 06 2020 *)

a(n) = A000010(A039771(n)). - Amiram Eldar, Mar 06 2020

A281016 Numbers k such that k, phi(k) and cototient(k) are all perfect powers.

Original entry on oeis.org

8, 16, 32, 64, 125, 128, 256, 512, 1024, 2048, 3125, 4096, 4913, 8192, 16384, 32768, 50653, 65536, 78125, 131072, 262144, 524288, 1030301, 1048576, 1419857, 1953125, 2097152, 4194304, 7645373, 8388608, 16777216, 16974593, 33554432, 35831808, 48828125, 64481201, 67108864, 69343957

Offset: 1

Altug Alkan, Jan 13 2017

nonn

This sequence does not contain only prime powers. Least term that has a prime factor which is not of the form m^2 + 1 is 35831808 = 2^14 * 3^7. The next one is 102503232 = 2^6 * 3^6 * 13^3. There are infinitely many such numbers.

Contains 2^a * 3^b whenever min(GCD(a,b), GCD(a,b-1), GCD(a+1,b-1)) > 1, e.g. if a == 14 (mod 42) and b == 7 (mod 42). - Robert Israel, Jul 08 2025

			125 = 5^3 is a term because phi(5^3) = 10^2 and cototient(5^3) = 5^2.

David A. Corneth, Table of n, a(n) for n = 1..547 (first 158 terms from Robert Israel, terms <= 10^20)

Cf. A000010, A001597, A051953, A054754, A166955.

ispow:= proc(n) local F;
  F:= ifactors(n)[2];
  igcd(F[..,2]) > 1
end proc:
N:= 10^8: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
Cands:= {seq(seq(i^p, i = 2 .. floor(N^(1/p))),p = P)}:
filter:= proc(n) local t; t:= numtheory:-phi(n); ispow(t) and ispow(n-t) end proc:
sort(convert(select(filter, Cands),list)); # Robert Israel, Jul 08 2025

Select[Range[10^6], Times @@ Boole@ Map[Or[# == 1, GCD @@ FactorInteger[#][[All, 2]] > 1] &, {#, EulerPhi@ #, # - EulerPhi@ #}] > 0 &] (* Michael De Vlieger, Jan 14 2017 *)

is(n) = ispower(eulerphi(n)) && ispower(n-eulerphi(n)) && ispower(n);

A281069 Least k such that phi(k) is an n-th power when k is the product of n distinct primes.

Original entry on oeis.org

2, 10, 30, 3458, 29526, 5437705, 91604415, 1190857395, 26535163830, 344957129790

Offset: 1

Altug Alkan, Jan 14 2017

Subsequence of A166955.
Corresponding values of phi(k) are 1, 4, 8, 1296, 7776, 2985984, ...
Freiberg (Theorem 1.2) shows that there are >> (n*x^(1/n))/(log x)^(n+2) such values of k up to x. He calls the set of such numbers B*(x;-1;n). In particular, a(n) exists for each n.
a(11) <= 5703406198237930, a(12) <= 178435136773443810, a(13) <= 4961806417984478790. - Daniel Suteu, Apr 04 2021

			a(4) = 3458 = 2 * 7 * 13 * 19 and phi(3458) = (2-1)*(7-1)*(13-1)*(19-1) = 6^4.

W. D. Banks and F. Luca, Power totients with almost primes, author's copy, Integers 11A (2011), 307-313.
Tristan Freiberg, Products of shifted primes simultaneously taking perfect power values, arXiv:1008.1978 [math.NT], 2010.
Tristan Freiberg and Carl Pomerance, A note on square totients, International Journal of Number Theory, Volume 11, Issue 08, December 2015.

Cf. A000010, A001597, A005117, A166955, A281140.

a(n) = {my(k=2); while (!(issquarefree(k) && (omega(k)==n) && (ispower(eulerphi(k),n))), k++); k;} \\ Michel Marcus, Jan 16 2017

from itertools import count
from math import isqrt, prod
from sympy import perfect_power, primefactors, primerange, integer_nthroot, primepi
def A281069(n):
    def squarefreealmostprimepi(n,k):
        if k==0: return int(n>=1)
        def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
        return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,k)) if k>1 else primepi(n))
    return next(k for k in (squarefreealmostprime(i,n) for i in count(1)) if (p:=perfect_power(prod(p-1 for p in primefactors(k)))) and p[1] == n) if n>1 else 2 # Chai Wah Wu, Sep 09 2024

a(9)-a(10) from Jinyuan Wang, Nov 08 2020

A216452 The fourth roots of the fourth powers arising in A078164.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 10, 8, 8, 8, 10, 8, 10, 8, 8, 8, 10, 10, 10, 12, 12, 12, 12, 10, 12

Offset: 1

V. Raman, Sep 07 2012

nonn

Cf. A039770, A039771, A039772, A062732, A078164, A166955.

A332008 Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).

Original entry on oeis.org

1, 15, 16, 63, 101, 125, 255, 256, 272, 504, 512, 513, 629, 679, 1358, 1359, 1728, 1970, 2047, 2222, 2509, 2834, 3458, 3705, 4094, 4095, 4400, 4577, 4616, 4913, 5403, 6817, 7295, 7956, 8729, 11667, 11672, 16132, 16384, 16523, 17507, 23085, 24198, 24564, 24624, 25220, 25601, 27216, 27384, 28564, 29444

Offset: 1

Antonio Roldán, Feb 04 2020

nonn

			phi(101) = 10^2, and phi(102) = 2^5.
phi(3458) = 6^4, and phi(3459) = 48^2.

Cf. A000010, A001597, A166955, A281016.

[1] cat [k:k in [3..30000]|IsPower(EulerPhi(k))  and IsPower(EulerPhi(k+1))]; // Marius A. Burtea, Feb 05 2020

perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[30000], And @@ perfectPowerQ /@ EulerPhi[# + {0, 1}] &] (* Amiram Eldar, Feb 04 2020 *)

v=[1]; for(i = 2, 30000, if(ispower(eulerphi(i)), if(ispower(eulerphi(i+1)), v = concat(v, i)))); v

cp's OEIS Frontend

A065528 Numbers n such that phi(n) is a nontrivial power b^c where b > 1 and c > 1.

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A216412 The cubes arising in A039771.

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A281016 Numbers k such that k, phi(k) and cototient(k) are all perfect powers.

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A281069 Least k such that phi(k) is an n-th power when k is the product of n distinct primes.

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A216452 The fourth roots of the fourth powers arising in A078164.

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A332008 Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).

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Magma

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