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A263720 Palindromic numbers such that the sum of the digits equals the number of divisors.

%I A263720 #38 Feb 16 2025 08:33:27
%S A263720 1,2,11,22,101,202,444,525,828,1111,2222,4884,5445,5775,12321,13431,
%T A263720 18081,21612,24642,26862,31213,44244,44844,51415,52425,56265,62426,
%U A263720 80008,86868,89298,99099,135531,162261,198891,217712,237732,301103,343343,480084,486684,512215,521125
%N A263720 Palindromic numbers such that the sum of the digits equals the number of divisors.
%C A263720 Subsequence of A002113.
%C A263720 A000005(a(n)) = A007953(a(n)).
%C A263720 The only known palindromic primes whose sum of digits equals the numbers of divisors (primes of the form 10^k + 1) are 2,11,101.
%H A263720 Chai Wah Wu, <a href="/A263720/b263720.txt">Table of n, a(n) for n = 1..10000</a>
%H A263720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>
%H A263720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>
%H A263720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>
%e A263720 a(3) = 11, 11 is the palindromic number, digitsum(11) = 1 + 1 = 2, sigma_0(11) = 2.
%t A263720 fQ[n_] := Block[{d = IntegerDigits@ n}, And[d == Reverse@ d, Total@ d == DivisorSigma[0, n]]]; Select[Range[2^19], fQ] (* _Michael De Vlieger_, Oct 27 2015 *)
%t A263720 Select[Range[600000],PalindromeQ[#]&&Total[IntegerDigits[#]] == DivisorSigma[ 0,#]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 28 2019 *)
%o A263720 (PARI) lista(nn) = {for(n=1, nn, my(d = digits(n)); if ((Vecrev(d) == d) && (numdiv(n) == sumdigits(n)), print1(n, ", ")););} \\ _Michel Marcus_, Oct 25 2015
%Y A263720 Cf. A002113, A057531.
%K A263720 nonn,base
%O A263720 1,2
%A A263720 _Ilya Gutkovskiy_, Oct 24 2015