A062981 -id:A062981 - cp's OEIS frontend

A036878 a(n) = p^(p-1) where p = prime(n).

2, 9, 625, 117649, 25937424601, 23298085122481, 48661191875666868481, 104127350297911241532841, 907846434775996175406740561329, 88540901833145211536614766025207452637361, 550618520345910837374536871905139185678862401

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Also the least refactorable number (A033950) that has the n-th prime as its least prime factor. - Robert G. Wilson v, Jun 28 2006

			5^(5-1) = 5^4 = 625.

Vincenzo Librandi, Table of n, a(n) for n = 1..77
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

These integers are refactorable -- i.e., the number of divisors divides the number itself, cf. A033950.
Subset of A062981. Subsequence of A000169.
Subsequence of A111134 and A246655.

[p^(p-1): p in PrimesUpTo(50)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime@n^(Prime@n - 1), {n, 10}] (* Robert G. Wilson v, Jun 28 2006 *)
#^(#-1)&/@Prime[Range[10]] (* Harvey P. Dale, Oct 23 2015 *)

a(n) = my(p=prime(n)); p^(p-1); \\ Michel Marcus, Jan 24 2019

A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

Original entry on oeis.org

1, 6, 30, 66, 102, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, 672, 690, 714, 840, 870, 924, 930, 966, 1122, 1320, 1410, 1428, 1518, 1590, 1638, 1722, 1770, 1890, 1932, 2130, 2226, 2280, 2310, 2346, 2370, 2670, 2730, 2760, 2838, 2970, 2982, 3102, 3162, 3210, 3360, 3444, 3486, 3498, 3570, 3720, 3780, 3948, 3990

Offset: 1

Ilya Gutkovskiy, Oct 19 2016

nonn

Intersection of A003601, A120736 and A145551.
Numbers k such that A000005(k)|A007955(k), A000203(k)|A007955(k) and A000005(k)| A000203(k).
Numbers k such that A000005(k)|A062981(k), A072861(k)|A062758(k) and A245656(k) = 1.

			a(2) = 6 because 6 has 4 divisors {1,2,3,6}, 1*2*3*6/4 = 9, 1*2*3*6/(1 + 2 + 3 + 6) = 3 and (1 + 2 + 3 + 6)/4 = 3 are integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..20000
Eric Weisstein's World of Mathematics, Divisor Product.
Eric Weisstein's World of Mathematics, Divisor Function.
Wikipedia, Arithmetic number.

Cf. A000005, A000203, A003601, A007955, A062758, A062981, A072861, A120736, A145551, A245656.

with(numtheory): P:=proc(q) local a,b,k,n;for n from 1 to q do
a:=divisors(n); b:=mul(a[k],k=1..nops(a));
if type(sigma(n)/tau(n),integer) and type(b/sigma(n),integer) and
type(b/tau(n),integer) then print(n); fi;
od; end: P(10^5); # Paolo P. Lava, Oct 20 2016

Select[Range[4000], Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[1, #1]] && Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[0, #1]] && Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] & ]

A066916 a(n) = n^phi(n) - 1.

Original entry on oeis.org

0, 1, 8, 15, 624, 35, 117648, 4095, 531440, 9999, 25937424600, 20735, 23298085122480, 7529535, 2562890624, 4294967295, 48661191875666868480, 34012223, 104127350297911241532840, 25599999999, 7355827511386640, 26559922791423, 907846434775996175406740561328, 110075314175

Offset: 1

Jason Earls, Jan 23 2002

nonn

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A000010, A062981, A066727, A066915.

Table[n^EulerPhi[n]-1,{n,30}] (* Harvey P. Dale, May 06 2015 *)

a(n) = { n^eulerphi(n) - 1 } \\ Harry J. Smith, Apr 06 2010

a(n) = A062981(n) - 1.

A066915 a(n) = n^phi(n) + 1.

Original entry on oeis.org

2, 3, 10, 17, 626, 37, 117650, 4097, 531442, 10001, 25937424602, 20737, 23298085122482, 7529537, 2562890626, 4294967297, 48661191875666868482, 34012225, 104127350297911241532842, 25600000001, 7355827511386642, 26559922791425, 907846434775996175406740561330, 110075314177

Offset: 1

Jason Earls, Jan 23 2002

nonn

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A000010, A062981, A066719, A066916.

Table[n^EulerPhi[n]+1,{n,30}] (* Harvey P. Dale, May 29 2014 *)

a(n) = { n^eulerphi(n) + 1 } \\ Harry J. Smith, Apr 06 2010

a(n) = A062981(n) + 1.

A175504 a(n) = n ^ (phi(n) - 1), phi(n) = A000010(n) = Euler totient function.

Original entry on oeis.org

1, 1, 3, 4, 125, 6, 16807, 512, 59049, 1000, 2357947691, 1728, 1792160394037, 537824, 170859375, 268435456, 2862423051509815793, 1889568, 5480386857784802185939, 1280000000, 350277500542221

Offset: 1

Jaroslav Krizek, May 31 2010

nonn

a(n) = A062981(n) / n. a(n) = A001783(n) / n * Product_{d|n} (d!/d^d)^A008683(n/d) = (Product_{GCD(k, n)=1} k) / (n * Product_{d|n} (d!/d^d)^A008683(n/d)).

Table[n^(EulerPhi[n]-1),{n,25}]  (* Harvey P. Dale, Feb 13 2011 *)

a(n)=n^(eulerphi(n)-1) \\ Charles R Greathouse IV, Mar 05 2013

A175701 a(n) = n ^ (phi(n)+1), phi(n) = A000010(n) = Euler totient function.

Original entry on oeis.org

1, 4, 27, 64, 3125, 216, 823543, 32768, 4782969, 100000, 285311670611, 248832, 302875106592253, 105413504, 38443359375, 68719476736, 827240261886336764177, 612220032, 1978419655660313589123979

Offset: 1

Jaroslav Krizek, Aug 09 2010

nonn

			For n = 6, a(6) = 6 ^ (phi(6)+1) = 6 ^ (A000010(6)+1) = 6 ^ (2+1) = 216.

Vincenzo Librandi, Table of n, a(n) for n = 1..300

[n^(EulerPhi(n)+1): n in [1..25]]; // Vincenzo Librandi, Jul 13 2012

A175701 := proc(n)
    n^(numtheory[phi](n)+1) ;
end proc: # R. J. Mathar, Apr 01 2014

Table[n^(EulerPhi[n]+1),{n,1,30}] (* Vincenzo Librandi, Jul 13 2012 *)

a(n)=n^(eulerphi(n)+1) \\ Charles R Greathouse IV, Mar 05 2013

a(n) = n*A062981(n). - R. J. Mathar, Apr 01 2014

A261768 a(n) = phi(n)^n - n^phi(n), where phi(n) is Euler's totient function.

Original entry on oeis.org

0, -1, -1, 0, 399, 28, 162287, 61440, 9546255, 1038576, 74062575399, 16756480, 83695120256591, 78356634560, 35181809198207, 281470681743360, 246486713303685957375, 101559922656192, 604107995057426434824791, 1152921479006846976

Offset: 1

Ilya Gutkovskiy, Aug 31 2015

sign

a(n) < n^n/e. If n is prime, a(n)/n^n = (1-1/n)^n - 1/n -> 1/e as n -> infinity. - Robert Israel, Sep 18 2015

Robert Israel, Table of n, a(n) for n = 1..388
Eric Weisstein's World of Mathematics, Totient Function

Cf. A000010, A062981.

[EulerPhi(n)^n-n^EulerPhi(n): n in [1..20]]; // Vincenzo Librandi, Sep 01 2015

seq(numtheory:-phi(n)^n - n^numtheory:-phi(n),n=1..30); # Robert Israel, Sep 18 2015

Table[EulerPhi[n]^n - n^EulerPhi[n], {n, 1, 20}]

a(n) = eulerphi(n)^n - n^eulerphi(n) \\ Anders Hellström, Aug 31 2015

a(n) = A000010(n)^n - n^A000010(n) = A000010(n)^n - A062981(n).

A284438 a(n) = phi(n)^n.

Original entry on oeis.org

1, 1, 8, 16, 1024, 64, 279936, 65536, 10077696, 1048576, 100000000000, 16777216, 106993205379072, 78364164096, 35184372088832, 281474976710656, 295147905179352825856, 101559956668416, 708235345355337676357632, 1152921504606846976, 46005119909369701466112, 10000000000000000000000

Offset: 1

Vincenzo Librandi, Apr 05 2017

nonn

			a(4) = 16 because phi(4)^4 = 2^4 = 16.
a(5) = 1024 because phi(5)^5 = 4^5 = 1024.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000010, A062981.

Magma
```
[EulerPhi(n)^n: n in [1..25]];
```
Mathematica
```
Table[EulerPhi[n]^n, {n, 30}]
```

cp's OEIS Frontend

A036878 a(n) = p^(p-1) where p = prime(n).

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A277521 Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.

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A066916 a(n) = n^phi(n) - 1.

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A066915 a(n) = n^phi(n) + 1.

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A175504 a(n) = n ^ (phi(n) - 1), phi(n) = A000010(n) = Euler totient function.

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A175701 a(n) = n ^ (phi(n)+1), phi(n) = A000010(n) = Euler totient function.

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A261768 a(n) = phi(n)^n - n^phi(n), where phi(n) is Euler's totient function.

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A284438 a(n) = phi(n)^n.

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