A069747 -id:A069747 - cp's OEIS frontend

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

Jaroslav Krizek, Mar 23 2019

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

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

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

Cf. A007955, A002113, A069747.
Includes A002385.
Similar sequences for functions sigma(m) and tau(m): A028986, A324988.

[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))]

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

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 *)

ispal(n) = my(d=digits(n)); Vecrev(d) == d;
isok(n) = ispal(n) && ispal(vecprod(divisors(n))); \\ Michel Marcus, Mar 23 2019

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

A324988 Palindromes whose number of divisors is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 262, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 424, 434, 454, 474, 484, 494, 505, 515, 535, 545

Offset: 1

Jaroslav Krizek, Mar 23 2019

Numbers m such that m and A000005(m) = tau(m) are both in A002113.

			Number of divisors of palindrome number 22 with divisors 1, 2, 11 and 22 is 4 (palindrome number).

Cf. A000005, A002113, A069747.
Similar sequences for functions sigma(m) and pod(m): A028986, A324989.
Includes A002385, A046328 and A046329.

[n: n in [1..1000] | Intseq(n, 10) eq Reverse(Intseq(n, 10)) and Intseq(NumberOfDivisors(n), 10) eq Reverse(Intseq(NumberOfDivisors(n), 10))]

ispali:= proc(n) local L; L:= convert(n,base,10); L = ListTools:-Reverse(L) end proc:
select(t -> ispali(t) and ispali(numtheory:-tau(t)), [$1..10000]); # Robert Israel, Mar 26 2019

Select[Range@ 600, And[PalindromeQ@ #, PalindromeQ@ DivisorSigma[0, #]] &] (* Michael De Vlieger, Mar 24 2019 *)

ispal(n) = my(d=digits(n)); Vecrev(d) == d;
isok(n) = ispal(n) && ispal(numdiv(n)); \\ Michel Marcus, Mar 23 2019

cp's OEIS Frontend

A324989 Palindromes whose product of divisors is palindromic.

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