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 A001248 Squares of primes. 319

%I

%S 4,9,25,49,121,169,289,361,529,841,961,1369,1681,1849,2209,2809,3481,

%T 3721,4489,5041,5329,6241,6889,7921,9409,10201,10609,11449,11881,

%U 12769,16129,17161,18769,19321,22201,22801,24649,26569,27889,29929,32041,32761,36481

%N Squares of primes.

%C Also 4, together with numbers n such that Sum_{d|n}(-1)^d = -A048272(n) = -3. - _Benoit Cloitre_, Apr 14 2002

%C Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1. - _Farideh Firoozbakht_, Feb 02 2005

%C Unique numbers having 3 divisors (1, their square root, themselves). - _Alexandre Wajnberg_, Jan 15 2006

%C Smallest (or first) new number deleted at the n-th step in an Eratosthenes sieve. - _Lekraj Beedassy_, Aug 17 2006

%C Subsequence of semiprimes A001358. - _Lekraj Beedassy_, Sep 06 2006

%C A000005(a(n)^(k-1)) = A005408(k) for all k>0. - _Reinhard Zumkeller_, Mar 04 2007

%C Integers having only 1 factor other than 1 and the number itself. Every number in the sequence is a multiple of 1 factor other than 1 and the number itself. 4 : 2 is the only factor other than 1 and 4; 9 : 3 is the only factor other than 1 and 9; and so on. - Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 23 2007

%C The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - _Omar E. Pol_, May 06 2008

%C There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no non-Abelian group. - _Franz Vrabec_, Sep 11 2008

%C For n > 2: (a(n) + 17) mod 12 = 6. - _Reinhard Zumkeller_, May 12 2010

%C A192134(A095874(a(n))) = A005722(n) + 1. - _Reinhard Zumkeller_, Jun 26 2011

%C For n > 2: a(n) = 1 (mod 24). - _Moshe Levin_, Dec 07 2011

%C A211110(a(n)) = 2. - _Reinhard Zumkeller_, Apr 02 2012

%C Solutions of the differential equation n'=2*sqrt(n), where n' is the arithmetic derivative of n. - _Paolo P. Lava_, Apr 23 2012

%C Also numbers n such that phi(n) = n - sqrt(n). - _Michel Lagneau_, May 25 2012

%C a(n) = A087112(n,n). - _Reinhard Zumkeller_, Nov 25 2012

%C For n > 1, n is the sum of numbers from A006254(n-1) to A168565(n-1). - _Vicente Izquierdo Gomez_, Dec 01 2012

%C Numbers whose multiplicative projection (A000026) is equal to their arithmetic derivative (A003415). - _Paolo P. Lava_, Dec 11 2012

%C A078898(a(n)) = 2. - _Reinhard Zumkeller_, Apr 06 2015

%H N. J. A. Sloane, <a href="/A001248/b001248.txt">Table of n, a(n) for n = 1..5000</a>

%H R. P. Boas & N. J. A. Sloane, <a href="/A005381/a005381.pdf">Correspondence, 1974</a>

%H M. Coman, <a href="https://www.researchgate.net/profile/Marius_Coman/publication/280626584">On the special relation between the numbers of the form 505+ 1008k and the squares of primes</a>, 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Index_entries_for_number_of_divisors">Index entries for number of divisors</a>

%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>

%F n such that A062799(n) = 2. - _Benoit Cloitre_, Apr 06 2002

%F a(n) = A000040(n)^(3-1)=A000040(n)^2, where 3 is the number of divisors of a(n). - _Omar E. Pol_, May 06 2008

%F A000005(a(n)) = 3 or A002033(a(n)) = 2. - _Juri-Stepan Gerasimov_, Oct 10 2009

%F A033273(a(n)) = 3. - _Juri-Stepan Gerasimov_, Dec 07 2009

%F a(n) = prime(n)^2. - _Jon E. Schoenfield_, Mar 29 2015

%F Product_{n>=1} a(n)/(a(n)-1) = Pi^2/6. - _Daniel Suteu_, Feb 06 2017

%p A001248:=n->ithprime(n)^2; seq(A001248(k), k=1..50); # _Wesley Ivan Hurt_, Oct 11 2013

%t Prime[Range[30]]^2 (* _Moshe Levin_, Dec 07 2011 *)

%t Select[Range[40000], DivisorSigma[0, #] == 3 &] (* _Carlos Eduardo Olivieri_, Jun 01 2015 *)

%o (Sage) BB = primes_first_n(36) list = [] for i in range(36): list.append(BB[i]^2) list # _Zerinvary Lajos_, May 15 2007

%o (PARI) forprime(p=2,1e3,print1(p^2", ")) \\ _Charles R Greathouse IV_, Jun 10 2011

%o (PARI) A001248(n)=prime(n)^2 \\ _M. F. Hasler_, Sep 16 2012

%o a001248 n = a001248_list !! (n-1)

%o a001248_list = map (^ 2) a000040_list -- _Reinhard Zumkeller_, Sep 23 2011

%o (MAGMA) [p^2: p in PrimesUpTo(300)]; // _Vincenzo Librandi_, Mar 27 2014

%Y Cf. A000040, A049001, A024450 (partial sums), A008864, A060800.

%Y Subsequence of A000430.

%Y Cf. A078898.

%Y Cf. A001694, A258599.

%Y Cf. A069482 (first differences).

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

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