A356750 Palindromic odd numbers with an odd number of distinct prime factors.

3, 5, 7, 9, 11, 101, 121, 131, 151, 181, 191, 313, 343, 353, 373, 383, 525, 555, 585, 595, 727, 757, 777, 787, 797, 919, 929, 969, 1001, 1221, 1331, 1551, 1771, 1881, 3333, 3553, 3663, 5225, 5335, 5445, 5555, 5665, 5885, 5995, 7007, 7227, 7337, 7557, 7667, 7777, 7887, 9339, 9669, 9779, 9889, 9999, 10201, 10301

Offset: 1

Author

Tanya Khovanova, Aug 25 2022

Keywords

• nonn
• base

Comments

Numbers in this sequence can be divided by nontrivial prime powers.

This sequence contains palindromic primes: A002385.

This sequence contains palindromic odd composite numbers that are products of an odd number of distinct primes: A075808.

Examples

			Number 525 = 3*5^2*7 has 3 prime factors 3, 5, and 7. Thus, it is in the sequence.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Programs

Mathematica

Select[Range[2,12000], OddQ[#] && PalindromeQ[#] && OddQ[Length[Transpose[FactorInteger[#]][[2]]]] &]

PARI

ispal(n) = my(d1=digits(n)); d1 == Vecrev(d1)
forstep(k=3,10^6,2,if(ispal(k)&&omega(k)%2==1,print1(k,", "))) \\ Alexandru Petrescu, Sep 10 2022

Python

from sympy import isprime, factorint
from itertools import count, islice, product
def cond(n): return n&1 and (isprime(n) or len(factorint(n))&1)
def oddpals(): # generator of odd palindromes
    yield from [1, 3, 5, 7, 9]
    for d in count(2):
        for first in "13579":
            for p in product("0123456789", repeat=(d-2)//2):
                left = "".join(p); right = left[::-1]
                for mid in [[""], "0123456789"][d%2]:
                    yield int(first + left + mid + right + first)
def agen(): yield from filter(cond, oddpals())
print(list(islice(agen(), 58))) # Michael S. Branicky, Aug 25 2022