OFFSET
1,1
COMMENTS
Numbers k such that bigomega(k)^(bigomega(k)) = k, where bigomega = A001222. - Lekraj Beedassy, Aug 21 2004
Positive k such that k' = k, where k' is the arithmetic derivative of k. - T. D. Noe, Oct 12 2004
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
Subsequence of A100716; A003415(m*a(n)) = A129283(m)*a(n), especially A003415(a(n)) = a(n). - Reinhard Zumkeller, Apr 07 2007
A168036(a(n)) = 0. - Reinhard Zumkeller, May 22 2015
REFERENCES
J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 740 pp. 95; 312, Ellipses Paris 2004.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..40
David Beckwith, Problem 11158, American Mathematical Monthly, Vol. 112, No. 5 (May 2005), p. 468.
Jurij Kovic, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences, Vol. 15 (2012), #12.3.8.
FORMULA
Sum_{n>=1} 1/a(n) = A094289. - Amiram Eldar, Oct 13 2020
EXAMPLE
a(1) = 2^2 = 4.
a(2) = 3^3 = 27.
a(3) = 5^5 = 3125.
MAPLE
MATHEMATICA
Array[Prime[ # ]^Prime[ # ] &, 12] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
#^#&/@Prime[Range[10]] (* Harvey P. Dale, May 17 2024 *)
PROG
(Haskell)
a051674_list = map (\p -> p ^ p) a000040_list
-- Reinhard Zumkeller, Jan 21 2012
(PARI) a(n)=n=prime(n); n^n \\ Charles R Greathouse IV, Mar 20 2013
(Magma) [p^p: p in PrimesUpTo(30)]; // Vincenzo Librandi, Mar 27 2014
(Python) from gmpy2 import mpz
[mpz(prime(n))**mpz(prime(n)) for n in range(1, 100)] # Chai Wah Wu, Jul 28 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved