A001248 Squares of primes.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also 4, together with numbers n such that Sum_{d|n}(-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 n-th 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 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

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

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(n-1) to A168565(n-1). - Vicente Izquierdo Gomez, Dec 01 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)
• Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.
• 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

Crossrefs

Cf. A000005, A000040, A001358, A001694, A002033, A005408, A005722, A006254.

Cf. A008864, A033273, A049001, A048272, A051709, A059956, A060800, A062799.

Cf. A078898, A082020, A085548, A087112, A095874, A168565, A192134, A211110.

Cf. A258599.

Subsequence of A000430, A001358, and of A190641.

Cf. A024450 (partial sums), A069482 (first differences).

Programs

Haskell

a001248 n = a001248_list !! (n-1)
a001248_list = map (^ 2) a000040_list -- Reinhard Zumkeller, Sep 23 2011

Magma

[p^2: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

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 *)

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

Python

from sympy import prime
def A001248(n): return prime(n)**2 # Chai Wah Wu, Aug 09 2024

SageMath

[n^2 for n in prime_range(1,301)] # G. C. Greubel, May 02 2024

Formula

n such that A062799(n) = 2. - Benoit Cloitre, Apr 06 2002

A000005(a(n)^(k-1)) = A005408(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007

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

A000005(a(n)) = 3 or A002033(a(n)) = 2. - Juri-Stepan Gerasimov, Oct 10 2009

A033273(a(n)) = 3. - Juri-Stepan 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)