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A051674 a(n) = prime(n)^prime(n). 109

%I

%S 4,27,3125,823543,285311670611,302875106592253,827240261886336764177,

%T 1978419655660313589123979,20880467999847912034355032910567,

%U 2567686153161211134561828214731016126483469

%N a(n) = prime(n)^prime(n).

%C n such that bigomega(n)^(bigomega(n)) = n, where bigomega = A001222. - _Lekraj Beedassy_, Aug 21 2004

%C Positive n such that n' = n, where n' is the arithmetic derivative of n. - _T. D. Noe_, Oct 12 2004

%C David Beckwith proposes (in the AMM reference): "Let n be a positive integer and let p be a prime number. Prove that (p^p) | n! implies that (p^(p + 1)) | n!". - _Jonathan Vos Post_, Feb 20 2006

%C Subsequence of A100716; A003415(m*a(n)) = A129283(m)*a(n), especially A003415(a(n)) = a(n). - _Reinhard Zumkeller_, Apr 07 2007

%C A168036(a(n)) = 0. - _Reinhard Zumkeller_, May 22 2015

%D J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 740 pp. 95; 312, Ellipses Paris 2004.

%H T. D. Noe, <a href="/A051674/b051674.txt">Table of n, a(n) for n = 1..40</a>

%H David Beckwith, <a href="http://www.jstor.org/stable/30037505">Problem 11158</a>, American Mathematical Monthly, Vol. 112, No. 5 (May 2005), p. 468.

%H Jurij Kovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kovic/kovic4.html">The Arithmetic Derivative and Antiderivative</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.3.8.

%F a(n) = A000312(A000040(n)). - _Altug Alkan_, Sep 01 2016

%F Sum_{n>=1} 1/a(n) = A094289. - _Amiram Eldar_, Oct 13 2020

%e a(1) = 2^2 = 4.

%e a(2) = 3^3 = 27.

%e a(3) = 5^5 = 3125.

%p A051674:=n->ithprime(n)^ithprime(n): seq(A051674(n), n=1..10); # _Wesley Ivan Hurt_, Jun 25 2016

%t Array[Prime[ # ]^Prime[ # ] &, 12] (* _Vladimir Joseph Stephan Orlovsky_, May 01 2008 *)

%o (Haskell)

%o a051674_list = map (\p -> p ^ p) a000040_list

%o -- _Reinhard Zumkeller_, Jan 21 2012

%o (PARI) a(n)=n=prime(n);n^n \\ _Charles R Greathouse IV_, Mar 20 2013

%o (MAGMA) [p^p: p in PrimesUpTo(30)]; // _Vincenzo Librandi_, Mar 27 2014

%o (Python) from gmpy2 import mpz

%o [mpz(prime(n))**mpz(prime(n)) for n in range(1,100)] # _Chai Wah Wu_, Jul 28 2014

%Y Cf. A000040, A000312, A003415 (arithmetic derivative of n), A129150, A129151, A129152, A048102, A072873 (multiplicative closure), A104126.

%Y Subsequence of A100717; A203908(a(n)) = 0.

%Y Subsequence of A097764.

%Y Cf. A168036, A094289 (decimal expansion of Sum(1/p^p)).

%K nonn,easy

%O 1,1

%A Asher Auel (asher.auel(AT)reed.edu)

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Last modified March 8 13:59 EST 2021. Contains 341949 sequences. (Running on oeis4.)