A013822 -id:A013822 - cp's OEIS frontend

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

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

A013824 a(n) = 2^(5*n + 3).

Original entry on oeis.org

8, 256, 8192, 262144, 8388608, 268435456, 8589934592, 274877906944, 8796093022208, 281474976710656, 9007199254740992, 288230376151711744, 9223372036854775808, 295147905179352825856, 9444732965739290427392, 302231454903657293676544, 9671406556917033397649408

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (32).

Cf. A000079 (2^n), A009976, A013822, A013823, A016885 (5*n+3).

[2^(5*n+3): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

2^(5*Range[0, 20] + 3) (* Paolo Xausa, Feb 21 2025 *)

a(n)=2^(5*n+3) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 24 2008: (Start)
a(n) = 32*a(n-1), n > 0; a(0)=8.
G.f.: 8/(1-32*x).
a(n) = 8*A009976(n). (End)
From Elmo R. Oliveira, Feb 20 2025: (Start)
E.g.f.: 8*exp(32*x).
a(n) = A000079(A016885(n)). (End)
a(n) = 2*A013823(n) = 4*A013822(n). - Paolo Xausa, Feb 21 2025

A013823 a(n) = 2^(5*n + 2).

Original entry on oeis.org

4, 128, 4096, 131072, 4194304, 134217728, 4294967296, 137438953472, 4398046511104, 140737488355328, 4503599627370496, 144115188075855872, 4611686018427387904, 147573952589676412928, 4722366482869645213696, 151115727451828646838272, 4835703278458516698824704

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (32).

Cf. A000079, A009976, A013822, A013824, A016873.

[2^(5*n+2): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

2^(5*Range[0, 20] + 2) (* Paolo Xausa, Feb 21 2025 *)

makelist(2^(5*n+2),n,0,20); /* Martin Ettl, Oct 21 2012 */

From Philippe Deléham, Nov 24 2008: (Start)
a(n) = 32*a(n-1); a(0)=4.
G.f.: 4/(1-32*x).
a(n) = 4*A009976(n). (End)
From Elmo R. Oliveira, Feb 20 2025: (Start)
E.g.f.: 4*exp(32*x).
a(n) = A000079(A016873(n)). (End)
a(n) = 2*A013822(n) = A013824(n)/2. - Paolo Xausa, Feb 21 2025

A013894 a(n) = 20^(5*n + 1).

Original entry on oeis.org

20, 64000000, 204800000000000, 655360000000000000000, 2097152000000000000000000000, 6710886400000000000000000000000000, 21474836480000000000000000000000000000000, 68719476736000000000000000000000000000000000000, 219902325555200000000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (3200000).

Subsequence of A009964.

Cf. A013822, A013854, A016861.

[20^(5*n+1): n in [0..10]]; // Vincenzo Librandi, May 27 2011

NestList[3200000*# &, 20, 10] (* Paolo Xausa, Jul 13 2025 *)

a(n) = 3200000*a(n-1), a(0)=20. - Vincenzo Librandi, May 27 2011
From Elmo R. Oliveira, Jul 11 2025: (Start)
G.f.: 20/(1-3200000*x).
E.g.f.: 20*exp(3200000*x).
a(n) = A013822(n)*A013854(n) = A009964(A016861(n)). (End)

A100629 a(n) = 2^(5*prime(n) + 1).

Original entry on oeis.org

2048, 65536, 67108864, 68719476736, 72057594037927936, 73786976294838206464, 77371252455336267181195264, 79228162514264337593543950336, 83076749736557242056487941267521536, 89202980794122492566142873090593446023921664

Offset: 1

Parthasarathy Nambi, Dec 02 2004

nonn

			a(1) = 2^(5*2 + 1) = 2048.

[2^(5*NthPrime(n) + 1): n in [1..20]]; // Vincenzo Librandi, Aug 25 2015

Do[Print[2^(5Prime[n] + 1)], {n, 1, 10}] (* Stefan Steinerberger, Feb 15 2006 *)
2^(5*Prime[Range[20]]+1) (* Harvey P. Dale, Oct 01 2024 *)

a(n) = A013822(prime(n)) = A013822(A000040(n)). - Michel Marcus, Aug 27 2015

More terms from Stefan Steinerberger, Feb 15 2006

a(10) from Vincenzo Librandi, Aug 27 2015

cp's OEIS Frontend

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

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A013824 a(n) = 2^(5*n + 3).

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A013823 a(n) = 2^(5*n + 2).

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A013894 a(n) = 20^(5*n + 1).

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A100629 a(n) = 2^(5*prime(n) + 1).

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