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

1,1

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A046389, A356750.

Sequence in context: A205888 A186529 A274901 * A204365 A283158 A043511

Adjacent sequences: A075805 A075806 A075807 * A075809 A075810 A075811

Keyword

nonn,base

Author

Jani Melik, Oct 13 2002

Extensions

Edited by Dean Hickerson, Oct 21 2002

Name edited by Tanya Khovanova, Aug 26 2022

Status

approved