A335645 Smallest palindrome with exactly n distinct prime factors.

1, 2, 6, 66, 858, 6006, 222222, 20522502, 244868442, 6172882716, 231645546132, 49795711759794, 2415957997595142, 495677121121776594, 22181673755737618122, 5521159517777159511255, 477552751050050157255774, 200345274602020206472543002

Offset: 0

Author

Michael S. Branicky, Oct 02 2020

Keywords

• nonn
• base
• hard

Comments

max{A002110(n), A076886(n), A239696(n)} <= a(n) <= A046399(n).

No more terms with less than 17 digits.

Next term: 10^16 <= a(13) <= 495677121121776594.

Examples

			a(3) = 66 because 66 is the smallest palindromic number with 3 distinct prime factors: 2*3*11.
		

Links

• Michael S. Branicky, Python program (translation of Suteu's PARI)

Crossrefs

Subsequence of A002113.

Cf. A002110, A046399, A076886, A239696.

Programs

PARI

omega_palindromes(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q); if(q == 5 && v%2 == 0, next); while(v <= B, if(j==1, if(v>=A && fromdigits(Vecrev(digits(v))) == v, listput(list, v)), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_palindromes(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 05 2023

Python

from sympy import factorint
def A335645(n):
  d = 1
  while True:
    half = (d+1)//2
    for left in range(10**(half-1), 10**half):
      strleft = str(left)
      if d%2 == 0:
        m = int(strleft + strleft[::-1])
      else:
        m = int(strleft + (strleft[:-1])[::-1])
      if len(factorint(m)) == n:
        return m
    d += 1
print([A335645(n) for n in range(8)]) # Michael S. Branicky, Oct 02 2020

Extensions

a(13) from Michael S. Branicky and David A. Corneth, Oct 03 2020

a(14) from David A. Corneth, Oct 03 2020

a(15) from Daniel Suteu, Feb 05 2023

a(16) from Michael S. Branicky, Feb 06 2023

a(17) from Michael S. Branicky, Feb 23 2023