

A001248


Squares of primes.


481



4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481
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OFFSET

1,1


COMMENTS

Also 4, together with numbers n such that Sum_{dn}(1)^d = A048272(n) = 3.  Benoit Cloitre, Apr 14 2002
Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1.  Farideh Firoozbakht, Feb 02 2005
Unique numbers having 3 divisors (1, their square root, themselves).  Alexandre Wajnberg, Jan 15 2006
Smallest (or first) new number deleted at the nth step in an Eratosthenes sieve.  Lekraj Beedassy, Aug 17 2006
Subsequence of semiprimes A001358.  Lekraj Beedassy, Sep 06 2006
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
The nth number with p divisors is equal to the nth prime raised to power p1, where p is prime.  Omar E. Pol, May 06 2008
There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no nonAbelian group.  Franz Vrabec, Sep 11 2008
Solutions of the differential equation n'=2*sqrt(n), where n' is the arithmetic derivative of n.  Paolo P. Lava, Apr 23 2012
Also numbers n such that phi(n) = n  sqrt(n).  Michel Lagneau, May 25 2012
For n > 1, n is the sum of numbers from A006254(n1) to A168565(n1).  Vicente Izquierdo Gomez, Dec 01 2012
Numbers whose multiplicative projection (A000026) is equal to their arithmetic derivative (A003415).  Paolo P. Lava, Dec 11 2012
A078898(a(n)) = 2.  Reinhard Zumkeller, Apr 06 2015
Let r(n) = (a(n)  1)/(a(n) + 1); then Product_{n>=1} r(n) = (3/5) * (4/5) * (12/13) * (24/25) * (60/61) * ... = 2/5.  Dimitris Valianatos, Feb 26 2019
Numbers k such that A051709(k) = 1.  Jianing Song, Jun 27 2021


LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from N. J. A. Sloane)
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Marius Coman, On the special relation between the numbers of the form 505+ 1008k and the squares of primes, 2015.
Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature


FORMULA

n such that A062799(n) = 2.  Benoit Cloitre, Apr 06 2002
A000005(a(n)^(k1)) = A005408(k) for all k>0.  Reinhard Zumkeller, Mar 04 2007
a(n) = A000040(n)^(31)=A000040(n)^2, where 3 is the number of divisors of a(n).  Omar E. Pol, May 06 2008
A000005(a(n)) = 3 or A002033(a(n)) = 2.  JuriStepan Gerasimov, Oct 10 2009
A033273(a(n)) = 3.  JuriStepan Gerasimov, Dec 07 2009
For n > 2: (a(n) + 17) mod 12 = 6.  Reinhard Zumkeller, May 12 2010
A192134(A095874(a(n))) = A005722(n) + 1.  Reinhard Zumkeller, Jun 26 2011
For n > 2: a(n) = 1 (mod 24).  Zak Seidov, Dec 07 2011
A211110(a(n)) = 2.  Reinhard Zumkeller, Apr 02 2012
a(n) = A087112(n,n).  Reinhard Zumkeller, Nov 25 2012
a(n) = prime(n)^2.  Jon E. Schoenfield, Mar 29 2015
Product_{n>=1} a(n)/(a(n)1) = Pi^2/6.  Daniel Suteu, Feb 06 2017
Sum_{n>=1} 1/a(n) = P(2) = 0.4522474200... (A085548).  Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(4) = 15/Pi^2 (A082020).
Product_{n>=1} (1  1/a(n)) = 1/zeta(2) = 6/Pi^2 (A059956). (End)


MAPLE

A001248:=n>ithprime(n)^2; seq(A001248(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013


MATHEMATICA

Prime[Range[30]]^2 (* Zak Seidov, Dec 07 2011 *)
Select[Range[40000], DivisorSigma[0, #] == 3 &] (* Carlos Eduardo Olivieri, Jun 01 2015 *)


PROG

(PARI) forprime(p=2, 1e3, print1(p^2", ")) \\ Charles R Greathouse IV, Jun 10 2011
(PARI) A001248(n)=prime(n)^2 \\ M. F. Hasler, Sep 16 2012
(Haskell)
a001248 n = a001248_list !! (n1)
a001248_list = map (^ 2) a000040_list  Reinhard Zumkeller, Sep 23 2011
(MAGMA) [p^2: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014


CROSSREFS

Cf. A000040, A049001, A024450 (partial sums), A008864, A059956, A060800, A082020, A085548.
Subsequence of A000430, A001358, and of A190641.
Cf. A078898, A001694, A258599.
Cf. A069482 (first differences).
Sequence in context: A179707 A247078 A077438 * A280076 A052043 A188836
Adjacent sequences: A001245 A001246 A001247 * A001249 A001250 A001251


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



