A039771 -id:A039771 - cp's OEIS frontend

A216412 The cubes arising in A039771.

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

A078164 Numbers k such that phi(k) is a perfect biquadrate.

Original entry on oeis.org

1, 2, 17, 32, 34, 40, 48, 60, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 1297, 1387, 1417, 1729, 1971, 2109, 2223, 2289, 2331, 2445, 2457, 2565, 2594, 2608, 2774, 2812, 2834, 2835, 3052, 3260, 3458, 3888, 3912, 3924, 3942, 3996, 4104, 4212, 4218

Offset: 1

Labos Elemer, Nov 27 2002

nonn

Corresponding values of phi include 1, 16, 256, 1296, 4096, ... and these arise several times each.
a(3) = A053576(4).
A013776 is a subsequence since phi(2^(4*n+1)) = (2^n)^4. - Bernard Schott, Sep 22 2022
Subsequence of primes is A037896 since in this case: phi(k^4+1) = k^4. - Bernard Schott, Mar 05 2023

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

Subsequence of A039770. A037896 is a subsequence.
Sequences where phi(k) is a perfect power: A039770 (square), A039771 (cube), this sequence (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th).
Cf. A001317, A053576, A037896, A045544, A000010, A013776.

k=4; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Select[Range[5000],IntegerQ[Surd[EulerPhi[#],4]]&] (* Harvey P. Dale, Apr 30 2015 *)

is(n)=ispower(eulerphi(n),4) \\ Charles R Greathouse IV, Apr 24 2020

from itertools import count, islice
from sympy import totient, integer_nthroot
def A078164_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:integer_nthroot(totient(n),4)[1], count(max(1,startvalue)))
A078164_list = list(islice(A078164_gen(),20)) # Chai Wah Wu, Feb 28 2023

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

A068559 Numbers m such that phi(m) = tau(m)^3.

Original entry on oeis.org

1, 85, 333, 436, 1542, 1875, 2907, 3285, 3488, 3796, 5196, 10280, 17532, 17776, 20080, 21250, 28305, 30368, 30555, 32708, 34748, 35308, 36860, 37060, 41544, 41568, 43068, 44004, 45162, 48468, 51930, 81324, 98304, 98688, 104856, 131070

Offset: 1

Benoit Cloitre, Mar 25 2002

For all large enough k, we have tau(k) < k^(1/4) and phi(k) > k^(3/4). Hence, tau(k)^3 < k^(3/4) < phi(k), implying that this sequence is finite. In fact, the sequence consists of 614 terms. - Max Alekseyev, May 30 2024

			a(2) = A107655(3) = 85.

Max Alekseyev, Table of n, a(n) for n = 1..614 (first 470 terms from Enrique Pérez Herrero)

Subsequence of A039771. - Enrique Pérez Herrero, Aug 29 2010

Cf. A020488, A068560, A107655, A114063.

Select[Range[132000],EulerPhi[#]==DivisorSigma[0,#]^3&]  (* Harvey P. Dale, Dec 28 2022 *)

isok(m) = eulerphi(m) == numdiv(m)^3; \\ Michel Marcus, Oct 18 2019

A236386 Numbers m such that phi(m) is an oblong number.

Original entry on oeis.org

3, 4, 6, 7, 9, 13, 14, 18, 21, 25, 26, 28, 31, 33, 36, 42, 43, 44, 49, 50, 62, 66, 73, 86, 87, 91, 95, 98, 111, 116, 117, 121, 135, 146, 148, 152, 157, 161, 169, 174, 182, 190, 201, 207, 211, 216, 222, 228, 234, 237, 241, 242, 252, 268, 270, 287, 289, 305

Offset: 1

Joseph L. Pe, Jan 24 2014

An oblong number (A002378) is of the form k*(k+1) where k is a natural number.
From Bernard Schott, Feb 27 2023: (Start)
Subsequence of primes is A002383 because in this case phi(k^2+k+1) = k*(k+1).
Subsequence of oblong numbers is A359847 where k and phi(k) are both oblong numbers. (End)

			phi(13) = 12 = 3*4, an oblong number; so 13 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000010, A002378, A002383, A359847.

Similar, but where phi(m) is: A039770 (square), A039771 (cube), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A360944 (triangular).

filter := m -> issqr(1 + 4*phi(m)) : select(filter, [$(1 .. 700)]); # Bernard Schott, Feb 26 2023

Select[Range[500], IntegerQ@Sqrt[1 + 4*EulerPhi[#]] &] (* Giovanni Resta, Jan 24 2014 *)

isok(m) = my(t=eulerphi(m)); !(t%2) && ispolygonal(t/2, 3); \\ Michel Marcus, Feb 27 2023

from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import totient
def A236386_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<2)+1), count(max(1,startvalue)))
A236386_list = list(islice(A236386_gen(),20)) # Chai Wah Wu, Feb 28 2023

a(16)-a(58) from Giovanni Resta, Jan 24 2014

cp's OEIS Frontend

A216412 The cubes arising in A039771.

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A078164 Numbers k such that phi(k) is a perfect biquadrate.

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

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

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

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