A324989 Palindromes whose product of divisors is palindromic.

1, 2, 3, 4, 5, 7, 11, 22, 101, 111, 121, 131, 151, 181, 191, 202, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 1001, 1111, 10001, 10201, 10301, 10501, 10601, 11111, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061

Offset

1,2

Comments

Numbers m such that m and A007955(m) = pod(m) are both in A002113.

Of 48025 terms < 10^11, all but 30 are prime. - Robert Israel, Apr 23 2019

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

Product of divisors of palindrome number 22 with divisors 1, 2, 11 and 22 is 484 (palindrome number).

Maple

revdigs:= proc(n)
local L, nL, i;
L:= convert(n, base, 10);
nL:= nops(L);
add(L[i]*10^(nL-i), i=1..nL);
end:
pals:= proc(d) local x, y;
  if d::even then [seq(x*10^(d/2)+revdigs(x), x=10^(d/2-1)..10^(d/2)-1)]
  else [seq(seq(x*10^((d+1)/2)+y*10^((d-1)/2)+revdigs(x), y=0..9), x=10^((d-1)/2-1)..10^((d-1)/2)-1)]
  fi
end proc:
pals(1):= [$1..9]:
filter:= proc(n) local v;
  v:= convert(numtheory:-divisors(n), `*`);
  revdigs(v)=v
end proc:
seq(op(select(filter, pals(d))), d=1..5); # Robert Israel, Apr 23 2019

Mathematica

Select[Range[10^5], And[PalindromeQ@ #, PalindromeQ[Times @@ Divisors@ #]] &] (* Michael De Vlieger, Mar 24 2019 *)
Select[Range[17000], AllTrue[{#, Times@@Divisors[#]}, PalindromeQ]&] (* Harvey P. Dale, Oct 13 2021 *)

Prog

(Magma) [n: n in [1..100000] | Intseq(n, 10) eq Reverse(Intseq(n, 10)) and Intseq(&*[d: d in Divisors(n)], 10) eq Reverse(Intseq(&*[d: d in Divisors(n)], 10))]
(PARI) ispal(n) = my(d=digits(n)); Vecrev(d) == d;
isok(n) = ispal(n) && ispal(vecprod(divisors(n))); \\ Michel Marcus, Mar 23 2019
(Python)
from math import isqrt
from itertools import chain, count, islice
from sympy import divisor_count
def A324989_gen(): # generator of terms
    return filter(lambda n:(s:=str(isqrt(n)**d if (d:=divisor_count(n)) & 1 else n**(d//2)))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1], chain.from_iterable(chain((int((s:=str(d))+s[-2::-1]) for d in range(10**l, 10**(l+1))), (int((s:=str(d))+s[::-1]) for d in range(10**l, 10**(l+1)))) for l in count(0)))
A324989_list = list(islice(A324989_gen(), 20)) # Chai Wah Wu, Jun 24 2022

Crossrefs

Cf. A007955, A002113, A069747.

Includes A002385.

Similar sequences for functions sigma(m) and tau(m): A028986, A324988.

Sequence in context: A279065 A327325 A340252 * A015856 A174165 A370946

Adjacent sequences: A324986 A324987 A324988 * A324990 A324991 A324992

Keyword

nonn,base

Author

Jaroslav Krizek, Mar 23 2019

Status

approved