A265640 Prime factorization palindromes (see comments for definition).

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 137, 139, 144

Offset: 1

Author

Vladimir Shevelev, Dec 11 2015

Keywords

• nonn

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

The sequence contains all perfect numbers except 6 (cf. A000396). - Don Reble, Dec 12 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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000396, A000720, A002113, A265641, complement of A229153.

Disjoint union of A229125 and (A000290\{0}).

Cf. A013661 (zeta(2)).

Programs

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 *)

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

Python

from math import isqrt
from sympy import primepi, mobius
def A265640(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n-(a:=isqrt(x))
        for y in range(1,a+1):
            m = x//y**2
            c -= primepi(m)-sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

Formula

lim A(x)/pi(x) = zeta(2) where A(x) is the number of a(n) <= x and pi is A000720.