A239696 -id:A239696 - cp's OEIS frontend

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

Michael S. Branicky, Oct 02 2020

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.

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

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

Subsequence of A002113.

Cf. A002110, A046399, A076886, A239696.

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

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

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

A113548 Least non-palindromic number k such that k and its digital reversal both have exactly n prime divisors.

Original entry on oeis.org

13, 12, 132, 1518, 15015, 204204, 10444434, 241879638, 20340535215, 242194868916, 136969856585562, 2400532020354468, 484576809394483806, 200939345091539746692

Offset: 1

Ryan Propper and Robert G. Wilson v, Sep 21 2005

This sequence does not allow ending in 0, else a(8) = 208888680, a(11) = 64635504163230 and a(13) = 477566276048801940. - Michael S. Branicky, Feb 14 2023

			a(1)=13=13 since 31=31,
a(2)=12=2^2*3 since 21=3*7,
a(3)=132=2^2*3*11 since 231=3*7*11,
...
a(7)=10444434=2*3*7*11*13*37*47 since 43444401=3*7*11*13*17*23*37,
a(8)=241879638=2*3*7*11*13*17*23*103 since 836978142=2*3*7*11*13*23*73*83.

Cf. A110843, A110819, A239696.

r[n_] := FromDigits[ Reverse[ IntegerDigits[ n]]]; f[n_] := Block[{k = r[n], len = Length[ FactorInteger[n]]}, If[k != n && len == Length[ FactorInteger[ r[n]]], len, 0]]; t = Table[0, {10}]; Do[ a = f[n]; If[a > 0 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 107}]; t

generate(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(q==5 && m%2==0, next); my(v=m*q); while(v <= B, if(j==1, my(r=fromdigits(Vecrev(digits(v)))); if(v>=A && r != v && omega(r) == n, 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) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 18 2023

a(n) >= A239696(n). - Daniel Suteu, Feb 18 2023

Edited and extended by Giovanni Resta, Jan 16 2006
a(9)-a(10) from Giovanni Resta, Feb 23 2014
a(11)-a(13) from Michael S. Branicky, Feb 14 2023
a(14) from Daniel Suteu, Feb 18 2023

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A335645 Smallest palindrome with exactly n distinct prime factors.

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