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

%I A075808 #26 Aug 28 2022 10:35:28
%S A075808 555,595,777,969,1001,1221,1551,1771,3333,3553,5335,5555,5665,5885,
%T A075808 5995,7337,7557,7667,7777,7887,9339,9669,9779,9889,11211,11811,12121,
%U A075808 12621,12921,13731,14241,14541,15051,15951,16261,16761,17171,18381
%N A075808 Palindromic odd composite numbers that are the products of an odd number of distinct primes.
%H A075808 Michael S. Branicky, <a href="/A075808/b075808.txt">Table of n, a(n) for n = 1..10000</a>
%e A075808 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.
%e A075808 50505 = 3 * 5 * 7 * 13 * 37 is the first term with five factors.
%e A075808 125 = 5^3 and 5445 = 3^2 * 5 * 11^2 are not terms since they are not the products of distinct primes.
%p A075808 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;
%t A075808 Select[Range[2,20000], ! PrimeQ[#] && OddQ[#] && PalindromeQ[#] &&
%t A075808    OddQ[Length[Transpose[FactorInteger[#]][[2]]]] &&
%t A075808 Max[Transpose[FactorInteger[#]][[2]]] == 1 &] (* _Tanya Khovanova_, Aug 26 2022 *)
%o A075808 (Python)
%o A075808 from sympy import isprime, factorint
%o A075808 from itertools import count, islice, product
%o A075808 def cond(n):
%o A075808     if n%2 == 0 or isprime(n): return False
%o A075808     f = factorint(n)
%o A075808     return len(f) == sum(f.values()) and len(f)&1
%o A075808 def oddpals(): # generator of odd palindromes
%o A075808     yield from [1, 3, 5, 7, 9]
%o A075808     for d in count(2):
%o A075808         for first in "13579":
%o A075808             for p in product("0123456789", repeat=(d-2)//2):
%o A075808                 left = "".join(p); right = left[::-1]
%o A075808                 for mid in [[""], "0123456789"][d%2]:
%o A075808                     yield int(first + left + mid + right + first)
%o A075808 def agen(): yield from filter(cond, oddpals())
%o A075808 print(list(islice(agen(), 38))) # _Michael S. Branicky_, Aug 25 2022
%Y A075808 Cf. A046389, A356750.
%K A075808 nonn,base
%O A075808 1,1
%A A075808 _Jani Melik_, Oct 13 2002
%E A075808 Edited by _Dean Hickerson_, Oct 21 2002
%E A075808 Name edited by _Tanya Khovanova_, Aug 26 2022