A075808 Palindromic odd composite numbers that are the products of an odd number of distinct primes.

555, 595, 777, 969, 1001, 1221, 1551, 1771, 3333, 3553, 5335, 5555, 5665, 5885, 5995, 7337, 7557, 7667, 7777, 7887, 9339, 9669, 9779, 9889, 11211, 11811, 12121, 12621, 12921, 13731, 14241, 14541, 15051, 15951, 16261, 16761, 17171, 18381

Offset: 1

Author

Jani Melik, Oct 13 2002

Keywords

• nonn
• base

Examples

			555 = 3*5*37, 595 = 5*7*17 and 777 = 3*7*37 are palindromic, odd, composite and products of an odd number of distinct primes.
50505 = 3 * 5 * 7 * 13 * 37 is the first term with five factors.
125 = 5^3 and 5445 = 3^2 * 5 * 11^2 are not terms since they are not the products of distinct primes.
		

Links

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

Crossrefs

Cf. A046389, A356750.

Programs

Maple

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[mobius](n)=-1 and not isprime(n); end; a := []; for n from 1 to 30000 by 2 do if test(n) then a := [op(a),n]; end; od; a;

Mathematica

Select[Range[2,20000], ! PrimeQ[#] && OddQ[#] && PalindromeQ[#] &&
   OddQ[Length[Transpose[FactorInteger[#]][[2]]]] &&
Max[Transpose[FactorInteger[#]][[2]]] == 1 &] (* Tanya Khovanova, Aug 26 2022 *)

Python

from sympy import isprime, factorint
from itertools import count, islice, product
def cond(n):
    if n%2 == 0 or isprime(n): return False
    f = factorint(n)
    return len(f) == sum(f.values()) and len(f)&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(), 38))) # Michael S. Branicky, Aug 25 2022

Extensions

Edited by Dean Hickerson, Oct 21 2002

Name edited by Tanya Khovanova, Aug 26 2022