A324988 -id:A324988 - 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

A327324 Palindromes whose number and sum of divisors are both also palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 333, 17571, 1757571, 1787871, 5136315, 518686815, 541626145, 17575757571, 5136813186315, 5136868686315, 5806270726085, 172757272757271, 513636363636315, 17275787578757271, 17578787578787571, 17878787578787871, 51363636363636315

Offset: 1

Jaroslav Krizek, Aug 30 2019

Numbers m such that m, A000005(m) = tau(m) and A000203(m) = sigma(m) are all in A002113.
Corresponding values of tau(a(n)): 1, 2, 2, 3, 2, 2, 6, 4, 4, 4, 8, 8, 8, 4, 8, 8, 8, 4, 8, ...
Corresponding values of sigma(a(n)): 1, 3, 4, 7, 6, 8, 494, 23432, 2343432, 2383832, ...
Intersection of A028986 and A324988.

			tau(333) = A000005(333) = 6; sigma(333) = A000203(333) = 494.

Cf. A000005, A000203, A002113, A028986, A324989, A324988.

[m: m in [1..1000000] | Intseq(m, 10) eq Reverse(Intseq(m, 10)) and Intseq(NumberOfDivisors(m), 10) eq Reverse(Intseq(NumberOfDivisors(m), 10)) and Intseq(&+[d: d in Divisors(m)], 10) eq Reverse(Intseq(&+[d: d in Divisors(m)], 10))];

Select[Range[2*10^6], PalindromeQ[#] && PalindromeQ[DivisorSigma[0, #]] && PalindromeQ[DivisorSigma[1, #]] &] (* Amiram Eldar, Aug 31 2019 *)

ispal(n) = my(d=digits(n)); d == Vecrev(d);
isok(n) = ispal(n) && ispal(numdiv(n)) && ispal(sigma(n)); \\ Michel Marcus, Sep 02 2019

a(20)-a(23) with the help of Daniel Suteu

cp's OEIS Frontend

A324989 Palindromes whose product of divisors is palindromic.

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