OFFSET
1,2
COMMENTS
a(66) is the first term at which this sequence differs from A119848.
A number N is called a prime factorization palindrome (PFP) if all its prime factors, taking into account their multiplicities, can be arranged in a row with central symmetry (see example). It is easy to see that every PFP-number is either a square or a product of a square and a prime. In particular, the sequence contains all primes.
Numbers which are both palindromes (A002113) and PFP are 1,2,3,4,5,7,9,11,44,99,101,... (see A265641).
If n is in the sequence, so is n^k for all k >= 0. - Altug Alkan, Dec 11 2015
Equivalently, numbers that have at most one prime factor with odd multiplicity. - Robert Israel, Feb 03 2016
Numbers whose squarefree part is noncomposite. - Peter Munn, Jul 01 2020
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
lim A(x)/pi(x) = zeta(2) where A(x) is the number of a(n) <= x and pi is A000720.
EXAMPLE
44 is a member, since 44=2*11*2.
52 is a member, since 52=2*13*2. [This illustrates the fact that the digits don't need to form a palindrome. This is not a base-dependent sequence. - N. J. A. Sloane, Oct 05 2024]
180 is a member, since 180=2*3*5*3*2.
MAPLE
N:= 1000: # to get all terms <= N
P:= [1, op(select(isprime, [2, seq(i, i=3..N, 2)]))]:
sort([seq(seq(p*x^2, x=1..floor(sqrt(N/p))), p=P)]); # Robert Israel, Feb 03 2016
MATHEMATICA
M = 200; P = Join[{1}, Select[Join[{2}, Range[3, M, 2]], PrimeQ]]; Sort[ Flatten[Table[Table[p x^2, {x, 1, Floor[Sqrt[M/p]]}], {p, P}]]] (* Jean-François Alcover, Apr 09 2019, after Robert Israel *)
PROG
(PARI) for(n=1, 200, if( ispseudoprime(core(n)) || issquare(n), print1(n, ", "))) \\ Altug Alkan, Dec 11 2015
(Python)
from math import isqrt
from sympy.ntheory.factor_ import core, isprime
def ok(n): return isqrt(n)**2 == n or isprime(core(n))
print([k for k in range(1, 145) if ok(k)]) # Michael S. Branicky, Oct 03 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Dec 11 2015
STATUS
approved