A005722 -id:A005722 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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).

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

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

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

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

forprime(p=2,1e3,print1(p^2", ")) \\ Charles R Greathouse IV, Jun 10 2011

A001248(n)=prime(n)^2  \\ M. F. Hasler, Sep 16 2012

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

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

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)

A192134 Difference between n-th prime power and its arithmetic derivative.

Original entry on oeis.org

1, 1, 2, 0, 4, 6, -4, 3, 10, 12, -16, 16, 18, 22, 15, 0, 28, 30, -48, 36, 40, 42, 46, 35, 52, 58, 60, -128, 66, 70, 72, 78, -27, 82, 88, 96, 100, 102, 106, 108, 112, 99, 50, 126, -320, 130, 136, 138, 148, 150, 156, 162, 166, 143, 172, 178, 180, 190, 192, 196

Offset: 1

Reinhard Zumkeller, Jun 26 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A000961, A001248, A003415, A005722, A006093, A025473, A025474, A095874, A192015, A192133.

a192134 n = a000961 n - a192015 n  -- Reinhard Zumkeller, Apr 16 2014

f[n_] := If[n == 1, 1, If[PrimePowerQ[n], {p, e} = FactorInteger[n][[1]]; n - e*p^(e-1), Nothing]]; Array[f, 300] (* Amiram Eldar, Apr 11 2025 *)

a(n) = A000961(n)-A192015(n) = A000961(n)-A003415(A000961(n)) = A192133(n)*A025473(n)^(A025474(n)-1) = A192133(n)*A000961(n)/A025473(n).
a(A095874(A000040(n))) = A006093(n).
a(A095874(A001248(n))) = A005722(n) + 1.

A173444 Either (n-th prime-1)^2-+1, but not both, is prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 12, 13, 19, 31, 32, 36, 37, 42, 47, 53, 54, 55, 58, 60, 63, 78, 79, 82, 83, 91, 94, 102, 105, 106, 118, 125, 126, 133, 135, 144, 155, 156, 159, 161, 163, 178, 184, 190, 206, 210, 214, 216, 219, 247, 248, 284, 286, 288, 307, 313, 315, 322, 336, 340, 344

Offset: 1

Juri-Stepan Gerasimov, Feb 18 2010, Mar 27 2010

nonn

Numbers n such that either A005722(n)-+1 is prime.

			1 is in the sequence because (1st prime-1)^2-1=0 is nonprime and (1st prime-1)^2+1=2 is prime;
3 is in the sequence because (3rd prime-1)^2-1=15 is nonprime and (3rd prime-1)^2+1=17 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A005722, A006093, A127435.

A005722 := proc(n) (ithprime(n)-1)^2 ; end proc: for n from 1 to 800 do a := A005722(n) ; if isprime(a-1) <> isprime(a+1) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 24 2010

ppQ[n_]:=Module[{c=(Prime[n]-1)^2},Sort[PrimeQ[{c+1,c-1}]]== {False, True}]; Select[Range[400],ppQ] (* Harvey P. Dale, Jun 24 2011 *)
Select[Range[400],Total[Boole[PrimeQ[(Prime[#]-1)^2+{1,-1}]]]==1&] (* Harvey P. Dale, Feb 01 2023 *)

More terms from R. J. Mathar, Apr 24 2010

Definition clarified by Harvey P. Dale, Jun 24 2011

A360283 a(n) = lcm({n! * binomial(n, k) for k = 0..n}).

Original entry on oeis.org

1, 1, 4, 18, 288, 1200, 43200, 529200, 11289600, 91445760, 9144576000, 92207808000, 13277924352000, 160283515392000, 2094371267788800, 58904191906560000, 15079473128079360000, 242109318556385280000, 78443419212268830720000, 1415903716781452394496000

Offset: 0

Peter Luschny, Feb 14 2023

nonn

Cf. A002397, A003418 (LCM), A005722, A021012, A196347.

a := n -> ilcm(seq(n!*binomial(n, k), k=0..n)):
seq(a(n), n = 0..19);

from math import factorial, lcm
def A360283(n): return factorial(n)*lcm(*(i for i in range(1,n+2)))//(n+1) # Chai Wah Wu, Feb 15 2023

a(n) = n! * lcm({k for k = 1..n+1}) / (n+1) = n! * LCM(n + 1) / (n + 1).

a(n) / a(n-1) = n^2 if and only if n + 1 is prime, for n >= 1.

A174165 Numbers n for which (prime(n) - 1)^2 +1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 12, 13, 19, 31, 32, 36, 37, 42, 47, 53, 54, 55, 58, 60, 63, 78, 79, 82, 83, 91, 94, 102, 105, 106, 118, 125, 126, 133, 135, 144, 155, 156, 159, 161, 163, 178, 184, 190, 206, 210, 214, 216, 219, 247, 248, 284, 286, 288, 307, 313, 315, 322

Offset: 1

Juri-Stepan Gerasimov, Mar 10 2010

nonn

			a(1) = 1 because (1st prime -1)^2 +1 = (2-1)^2 +1 = 2, a prime; a(2) = 2 because (2nd prime -1)^2 +1 = (3-1)^2 +1 = 5, a prime; a(3) = 3 because (3rd prime -1)^2 +1 = (5-1)^2 +1 = 17, a prime; ... ; a(6) # 6 since (6th prime -1)^2 = (13-1)^2 +1 = 145 = 5*29, which is not a prime; etc.

Cf. A000040, A006093, A005722 & A127435.

fQ[n_] := PrimeQ[ (Prime@n - 1)^2 + 1]; Select[ Range@ 335, fQ@# &]

Edited, corrected and extended by Robert G. Wilson v, Mar 14 2010

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A001248 Squares of primes.

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A192134 Difference between n-th prime power and its arithmetic derivative.

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A173444 Either (n-th prime-1)^2-+1, but not both, is prime.

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A360283 a(n) = lcm({n! * binomial(n, k) for k = 0..n}).

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A174165 Numbers n for which (prime(n) - 1)^2 +1 is prime.

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