A078164 -id:A078164 - cp's OEIS frontend

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

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.

A039770 Numbers k such that phi(k) is a perfect square.

Original entry on oeis.org

1, 2, 5, 8, 10, 12, 17, 32, 34, 37, 40, 48, 57, 60, 63, 74, 76, 85, 101, 108, 114, 125, 126, 128, 136, 160, 170, 185, 192, 197, 202, 204, 219, 240, 250, 257, 273, 285, 292, 296, 304, 315, 364, 370, 380, 394, 401, 432, 438, 444, 451, 456, 468, 489, 504, 505

Offset: 1

Olivier Gérard

A004171 is a subsequence because phi(2^(2k+1)) = (2^k)^2. - Enrique Pérez Herrero, Aug 25 2011
Subsequence of primes is A002496 since in this case phi(k^2+1) = k^2. - Bernard Schott, Mar 06 2023
Products of distinct terms of A002496 form a subsequence. - Chai Wah Wu, Aug 22 2025

			phi(34) = 16 = 4*4.

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 141.

T. D. Noe, Table of n, a(n) for n = 1..10000
W. D. Banks, J. B. Friedlander, C. Pomerance and I. E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.
P. Pollack and C. Pomerance, Square values of Euler's function, submitted for publication.
Bernard Schott, Subfamilies and subsequences

Cf. A000010, A007614. A062732 gives the squares. A306882 (squares not totient).
Cf. A068560, A114063, A078164.
Subsequences: A054755, A002496, A004171, A262406, A324745, A324746, A247129, A324747, A306908.

with(numtheory); isA039770 := proc (n) return issqr(phi(n)) end proc; seq(`if`(isA039770(n), n, NULL), n = 1 .. 505); # Nathaniel Johnston, Oct 09 2013

Select[ Range[ 600 ], IntegerQ[ Sqrt[ EulerPhi[ # ] ] ]& ]

for(n=1, 120, if (issquare(eulerphi(n)), print1(n, ", ")))

from math import isqrt
from sympy import totient as phi
def ok(n): return isqrt(p:=phi(n))**2 == p
print([k for k in range(1, 506) if ok(k)]) # Michael S. Branicky, Aug 17 2025

a(n) seems to be asymptotic to c*n^(3/2) with 1 < c < 1.3. - Benoit Cloitre, Sep 08 2002

Banks, Friedlander, Pomerance, and Shparlinski show that a(n) = O(n^1.421). - Charles R Greathouse IV, Aug 24 2009

A037896 Primes of the form k^4 + 1.

Original entry on oeis.org

2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001, 723394817, 916636177, 1049760001, 1416468497

Offset: 1

Donald S. McDonald, Feb 27 2000

From Bernard Schott, Apr 22 2019: (Start)
These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a perfect biquadrate: A307690, so this sequence is a subsequence of A078164 and A307690.
If p prime = k^4 + 1, phi(p) = k^4.
The last three Fermat primes in A019434 {17, 257, 65537} belong to this sequence; with F_k = 2^(2^k) + 1 and for k = 2, 3, 4, phi(F_k) = (2^(2^(k-2)))^4. (End)

			6^4 + 1 = 1297 is prime.

T. D. Noe, Table of n, a(n) for n=1..1000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
M. Lal, Primes of the form n^4 + 1, Math. Comp. 21 (1967), pp. 245-247.

Cf. A000068, A002523, A019434, A078164, A307690.

Subsequence of A002496, A078164 and A039770.

[n^4+1: n in [1..200] | IsPrime(n^4+1)]; // G. C. Greubel, Apr 28 2019

Select[Range[200]^4+1,PrimeQ] (* Harvey P. Dale, Jul 20 2015 *)

j=[]; for(n=1,200, if(isprime(n^4+1),j=concat(j,n^4+1))); j

list(lim)=my(v=List([2]),p); forstep(k=2,sqrtnint(lim\1-1,4),2, if(isprime(p=k^4+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 31 2022

[n^4+1 for n in (1..200) if is_prime(n^4+1)] # G. C. Greubel, Apr 28 2019

a(n) = A002523(A000068(n)). - Elmo R. Oliveira, Feb 21 2025

Corrected and extended by Jason Earls, Jul 19 2001

A013776 a(n) = 2^(4*n+1).

Original entry on oeis.org

2, 32, 512, 8192, 131072, 2097152, 33554432, 536870912, 8589934592, 137438953472, 2199023255552, 35184372088832, 562949953421312, 9007199254740992, 144115188075855872, 2305843009213693952

Offset: 0

N. J. A. Sloane

a(n) ~ -Pi*E(2*n)/B(2*n), E(n) Euler number, B(n) Bernoulli number. - Peter Luschny, Oct 28 2012
Equivalently, powers of 2 with final digit 2. - Muniru A Asiru, Mar 15 2019
As phi(a(n)) = (2^n)^4 is a perfect biquadrate (where phi is the Euler totient A000010), this is a subsequence of A078164 and A307690. - Bernard Schott, Mar 28 2022

			G.f. = 2 + 32*x + 512*x^2 + 8192*x^3 + 131072*x^4 + 2097152*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (16).

Subsequence of A307690.
Cf. A000010, A004171, A016813, A264960, A307690.
Intersection of A000079 and A078164.

List([0..20],n->2^(4*n+1)); # Muniru A Asiru, Mar 15 2019

[2^(4*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 27 2011

[2^(4*n+1)$n=0..20]; # Muniru A Asiru, Apr 10 2019

2^(4*Range[0,20]+1) (* G. C. Greubel, Mar 15 2019 *)
NestList[16#&,2,20] (* Harvey P. Dale, Jul 28 2019 *)

a(n)=2<<(4*n) \\ Charles R Greathouse IV, Apr 07 2012

[2^(4*n+1) for n in (0..20)] # G. C. Greubel, Mar 15 2019

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 16*a(n-1), n > 0, a(0) = 2.
G.f.: 2/(1 - 16*x). (End)
From Peter Bala, Nov 29 2015: (Start)
a(n) = Sum_{k = 0..n} binomial(2*k,k)*binomial(4*n + 2 - 2*k, 2*n + 1 - k).
Bisection of A264960. (End)
a(n) = A000079(A016813(n)). - Michel Marcus, Nov 30 2015
a(n) = Sum_{k = 0..2*n} binomial(4*n + 2, 2*k + 1) = A004171(2*n). - Peter Bala, Nov 25 2016
E.g.f.: 2*exp(16*x). - G. C. Greubel, Jun 30 2019
From Bernard Schott, Apr 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 8/15.
Sum_{n>=0} (-1)^n/a(n) = 8/17. (End)

Wrong comment deleted by Kevin Ryde, Apr 16 2022

A078165 Numbers k such that phi(k) is a perfect 5th power.

Original entry on oeis.org

1, 2, 51, 64, 68, 80, 96, 102, 120, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 7957, 8227, 8279, 9079, 9139, 9709, 9919, 10355, 10595, 11667, 11673, 11691, 12099, 12393, 12483, 12753, 12987, 13797, 14715, 14763

Offset: 1

Labos Elemer, Nov 27 2002

nonn

As phi(2^(5*n+1)) = (2^n)^5, A013822 is a subsequence. - Bernard Schott, Sep 26 2022

Numbers of the form u = 2^(5*k)*3^(5*m + 1), k>=1, m>=0, are terms because phi(u) = 2^(5*k)*3^(5*m) = (2^k*3^m)^5. - Marius A. Burtea, Sep 26 2022

			phi of the sequence includes 1, 32, 1024, 7776, ...; powers arise several times; a(3) = A053576(5) = 51.

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

A013822 is a subsequence.

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th, this sequence), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power), A001317, A053576, A045544, A000010.

k=5; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Select[Range[15000],IntegerQ[Surd[EulerPhi[#],5]]&] (* Harvey P. Dale, Jul 26 2019 *)

is(n)=ispower(eulerphi(n),5) \\ Charles R Greathouse IV, Apr 24 2020

A078166 Numbers k such that phi(k) is a perfect sixth power.

Original entry on oeis.org

1, 2, 85, 128, 136, 160, 170, 192, 204, 240, 4369, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 15360, 15420, 16320, 47197, 47239, 47989, 49267, 49589, 50557, 51319, 52429, 52649, 55699, 57589, 57953, 59495, 63973

Offset: 1

Labos Elemer, Nov 27 2002

nonn

As phi(2^(6*n+1)) = (2^n)^6, A277757 is a subsequence. - Bernard Schott, Sep 23 2022

			phi of the sequence includes 1, 64, 4096, 46656,..; powers arise several times; a(3)= A053576(6) = 85; in sequence relatively large jumps are observable when power of new numbers appear.

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

A277757 is a subsequence.
Numbers k such that phi(k) is a perfect power: A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th, this sequence), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power).
Cf. A001317, A053576, A045544, A000010.

k=6; Select[Range[65000],IntegerQ[EulerPhi[#]^(1/k)]&]  (* Harvey P. Dale, Feb 20 2011 *)

is(n)=ispower(eulerphi(n),6) \\ Charles R Greathouse IV, Apr 24 2020

A078167 Numbers k such that phi(k) is a perfect 7th power.

Original entry on oeis.org

1, 2, 255, 256, 272, 320, 340, 384, 408, 480, 510, 21845, 32768, 32896, 34816, 34952, 40960, 41120, 43520, 43690, 49152, 49344, 52224, 52428, 61440, 61680, 65280, 280999, 281587, 282637, 282949, 283897, 294409, 297449, 300409, 302039, 304399

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 128, 16384, 279936, etc..; powers arise several times; a(3) = A053576(7) = 255; in sequence rather large jumps arise when power of new numbers appear.

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

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th, this sequence), A078168 (8th), A078169 (9th), A078170 (10th power), A001317, A053576, A045544, A000010.

k=7; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 1000000}]

is(n)=ispower(eulerphi(n),7) \\ Charles R Greathouse IV, Apr 24 2020

A078168 Numbers k such that phi(k) is a perfect 8th power.

Original entry on oeis.org

1, 2, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 65537, 131072, 131074, 131584, 139264, 139808, 163840, 164480, 174080, 174760, 196608, 197376, 208896, 209712, 245760, 246720, 261120, 262140, 1682227, 1683109, 1683559, 1683937

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 256, 65536, 1679616, etc.; powers arise several times; a(3) = A053576(7) = 257; in sequence smoother ranges and quite large jumps arise when power of new numbers appear as phi-values.

Amiram Eldar, Table of n, a(n) for n = 1..7500

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th, this sequence), A078169 (9th), A078170 (10th power), A001317, A053576, A045544, A000010.

k=8; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]
Select[Range[2*10^6],IntegerQ[Surd[EulerPhi[#],8]]&] (* Harvey P. Dale, Oct 20 2014 *)

is(n)=ispower(eulerphi(n),8) \\ Charles R Greathouse IV, Apr 24 2020

A078169 Numbers k such that phi(k) is a perfect 9th power.

Original entry on oeis.org

1, 2, 771, 1024, 1028, 1088, 1280, 1360, 1536, 1542, 1632, 1920, 2040, 327685, 524288, 524296, 526336, 557056, 559232, 655360, 655370, 657920, 696320, 699040, 786432, 786444, 789504, 835584, 838848, 983040, 986880, 1044480, 1048560

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 512, 262144,.. etc.; powers arise several times; a(3) = A053576(9) = 771; in sequence smoother ranges and quite large jumps arise when power of new numbers appear as phi-values.

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

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th, this sequence), A078170 (10th power), A001317, A053576, A045544, A000010.

k=9; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]

is(n)=ispower(eulerphi(n),9) \\ Charles R Greathouse IV, Apr 24 2020

A078170 Numbers k such that phi(k) is a perfect tenth power.

Original entry on oeis.org

1, 2, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 1114129, 2097152, 2097184, 2105344, 2228224, 2228258, 2236928, 2621440, 2621480, 2631680, 2785280, 2796160, 3145728, 3145776, 3158016, 3342336

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 1024, 1048576,.. etc.; powers emerge several times; a(3) = A053576(10) = 1285; in sequence smoother ranges and quite large jumps alternate when power of new numbers appear as phi-values.

Amiram Eldar, Table of n, a(n) for n = 1..2250

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power, this sequence), A001317, A053576, A045544, A000010.

k=10; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]

is(n)=ispower(eulerphi(n),10) \\ Charles R Greathouse IV, Apr 24 2020

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

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A039770 Numbers k such that phi(k) is a perfect square.

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A037896 Primes of the form k^4 + 1.

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A013776 a(n) = 2^(4*n+1).

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A078165 Numbers k such that phi(k) is a perfect 5th power.

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A078166 Numbers k such that phi(k) is a perfect sixth power.

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A078167 Numbers k such that phi(k) is a perfect 7th power.

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