A078164 -id:A078164 - cp's OEIS frontend

A114063 Numbers k such that phi(k) = tau(k)^4, where tau(k) = A000005(k).

1, 17, 514, 8738, 32301, 33003, 36351, 41504, 42292, 43852, 51860, 62226, 549117, 561051, 571311, 599067, 617967, 629811, 634005, 657495, 673184, 674505, 683168, 701024, 705568, 718964, 722684, 732628, 745484, 759772, 774368

Offset: 1

Giovanni Resta, Feb 13 2006

For all large enough k, we have tau(k) < k^(1/5) and phi(k) > k^(4/5). Hence, tau(k)^4 < k^(4/5) < phi(k), implying that this sequence is finite. - Max Alekseyev, Mar 10 2016

Sequence is composed of 94030 terms. - Max Alekseyev, Jun 01 2024

			phi(33003) = 20736. tau(33003) = 12, 20736 = 12^4.
a(2) = A107655(4) = 17.

Max Alekseyev, Table of n, a(n) for n = 1..94030 (first 660 terms from Enrique Pérez Herrero; terms 661..7000 from Giovanni Resta)

Subsequence of A078164.

Cf. A020488, A062516, A063469, A063470, A107655, A112954, A112955, A175667, A068560, A068559.

Select[Range[10^6], EulerPhi[#] == DivisorSigma[0, #]^4 &] (* Paolo Xausa, May 31 2024 *)

isok(n) = eulerphi(n) == numdiv(n)^4; \\ Michel Marcus, Jan 22 2014

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

A307690 Integers with only one prime factor and whose Euler's totient is a perfect biquadrate.

Original entry on oeis.org

2, 17, 32, 257, 512, 1297, 8192, 65537, 131072, 160001, 331777, 614657, 1336337, 1419857, 2097152, 4477457, 5308417, 8503057, 9834497, 29986577, 33554432, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 536870912, 562448657, 655360001

Offset: 1

Bernard Schott, Apr 22 2019

nonn

An integer q is a term iff q = p^(4*m+1), when p is prime of the form k^4 + 1 and m >= 0, then phi(q) = (k * (k^4+1)^m)^4. The primitive terms of this sequence are the primes of the form p = k^4 + 1, which are exactly in A037896.

			a(14) = 1419857 = 17^5 and phi(1419857) = 34^4.

Subsequences: A013776 (2^(4*m+1)), A013806 (17^(4*m+1)), A037896 (primes of the form k^4 + 1).
Intersection of A078164 and A246655.
Cf. A054755 (idem with Euler's totient is square).

[n:n in [1..10000000]| #PrimeDivisors(n) eq 1 and IsPower(EulerPhi(n),4)]; // Marius A. Burtea, May 09 2019

isok(n) = isprimepower(n) && ispower(eulerphi(n), 4); \\ Michel Marcus, Apr 23 2019

A013806 a(n) = 17^(4*n+1).

Original entry on oeis.org

17, 1419857, 118587876497, 9904578032905937, 827240261886336764177, 69091933913008732880827217, 5770627412348402378939569991057, 481968572106750915091411825223071697, 40254497110927943179349807054456171205137

Offset: 0

N. J. A. Sloane

As phi(a(n)) = (2*17^n)^4 is a perfect biquadrate (where phi is the Euler totient A000010), this is a subsequence of A078164 and A307690. - Bernard Schott, Mar 29 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (83521).

Cf. A000010, A013776, A307690.

Intersection of A001026 and A078164.

[17^(4*n+1): n in [0..15]]; // Vincenzo Librandi, Jul 06 2011

17^(4Range[0,10]+1) (* or *) NestList[83521#&,17,20] (* Harvey P. Dale, May 21 2013 *)

a(0)=17, a(n)=83521*a(n-1). - Harvey P. Dale, May 21 2013
Sum_{n>=0} 1/a(n) = 4913/83520. - Bernard Schott, Mar 29 2022
Sum_{n>=0} (-1)^n/a(n) = 4913/83522. - Bernard Schott, Apr 08 2022

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

A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 16, 15, 8, 703, 247, 5, 247, 489, 1255, 5, 109, 247, 4, 3, 1, 247, 73, 3, 109, 1255, 13315, 163, 753, 109, 73, 109, 1373, 163, 27331, 1, 625, 1, 81, 109, 57, 73, 1295, 1, 251, 109, 74663, 625, 949, 13315, 1557377, 1, 74663, 753, 16, 81, 175765, 73, 251, 81, 37, 1373, 243895, 1

Offset: 1

Zhi-Wei Sun, Apr 24 2020

nonn

Conjecture: For any positive integers k and m, there is a positive integer n relatively prime to m such that phi(m*n) = phi(m)*phi(n) is a k-th power.
This conjecture implies that a(n) exists for every n = 1,2,3,....
See also A334353 for a similar conjecture involving the sigma function (A000203).
a(n) = 1 if and only if n is in A078164. - Charles R Greathouse IV, Apr 24 2020

			a(3) = 16 with gcd(3,16) = 1 and phi(3*16) = phi(3)*phi(16) = 2*8 = 2^4.
a(167) = 370517977 with gcd(167, 370517977) = 1 and phi(167*370517977) = phi(167)*phi(370517977) = 166*370517976 = 61505984016 = 498^4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..226
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000010, A000203, A000583, A039770, A039771, A259915, A259916, A334337, A280988, A334339, A334353, A078164.

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
phi[n_]:=phi[n]=EulerPhi[n];
tab={};Do[m=0;Label[aa];m=m+1;If[GCD[m,n]==1&&QQ[phi[m]*phi[n]],tab=Append[tab,m],Goto[aa]],{n,1,60}];tab

a(n) = my(m=1,e=eulerphi(n)); while (!((gcd(n, m) == 1) && ispower(e*eulerphi(m), 4)), m++); m; \\ Michel Marcus, Apr 25 2020

A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 7, 9, 11, 14, 18, 22, 29, 37, 57, 58, 63, 67, 74, 76, 79, 108, 114, 126, 134, 137, 143, 155, 158, 175, 183, 191, 211, 225, 231, 244, 248, 274, 277, 286, 308, 310, 329, 341, 350, 366, 372, 379, 382, 396, 417, 422, 423, 450, 453, 462, 554, 556, 604, 623, 631, 658, 682

Offset: 1

Bernard Schott, Feb 26 2023

nonn

Subsequence of primes is A055469 because in this case phi(k(k+1)/2+1) = k(k+1)/2.

Subsequence of triangular numbers is A287472.

			phi(57) = 36 = 8*9/2, a triangular number; so 57 is a term of the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000217, A055469, A287472.

Similar, but with phi(m) is: A039770 (square), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A236386 (oblong).

filter := m ->  issqr(1 + 8*numtheory:-phi(m)) : select(filter, [$(1 .. 700)]);

Select[Range[700], IntegerQ[Sqrt[8 * EulerPhi[#] + 1]] &] (* Amiram Eldar, Feb 27 2023 *)

isok(m) = ispolygonal(eulerphi(m), 3); \\ Michel Marcus, Feb 27 2023

from itertools import islice, count
from sympy.ntheory.primetest import is_square
from sympy import totient
def A360944_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<3)+1), count(max(1,startvalue)))
A360944_list = list(islice(A360944_gen(),20)) # Chai Wah Wu, Feb 28 2023

A114573 Numbers k such that phi(k) is a perfect 11th power.

Original entry on oeis.org

1, 2, 3855, 4096, 4112, 4352, 5120, 5140, 5440, 6144, 6168, 6528, 7680, 7710, 8160, 5570645, 8388608, 8388736, 8421376, 8912896, 8913032, 8947712, 10485760, 10485920, 10526720, 11141120, 11141290, 11184640, 12582912, 12583104

Offset: 1

Stefan Steinerberger, Feb 17 2006

nonn

Given the fact that phi(n) > sqrt(n) for all n except n=2 and n=6 we can see that every 11th power does appear as value only a finite number of times. What bounds on the density of this sequence can be proved?

			phi(4096) = 2048 = 2^11.

Amiram Eldar, Table of n, a(n) for n = 1..3625

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

For[n = 1, n < 100000, n++, If[EulerPhi[n]^(1/11) == Floor[EulerPhi[n]^(1/11)], Print[n]]]

More terms from Stefan Steinerberger, May 16 2007

cp's OEIS Frontend

A114063 Numbers k such that phi(k) = tau(k)^4, where tau(k) = A000005(k).

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

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A307690 Integers with only one prime factor and whose Euler's totient is a perfect biquadrate.

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A013806 a(n) = 17^(4*n+1).

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A216412 The cubes arising in A039771.

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A334350 Least positive integer m relatively prime to n such that phi(m*n) = phi(m)*phi(n) is a fourth power, where phi is Euler's totient function (A000010).

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A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

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A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).