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A309654 Numbers k such that k multiplied by the sum of reciprocals of digits is the digit reversal of k.

%I A309654 #23 Sep 08 2022 08:46:21
%S A309654 1,22,333,424,864,3663,4444,6336,39993,46664,48484,55555,64646,66366,
%T A309654 84448,88288,93939,362436,488884,666666,848848,884488,6699966,6886886,
%U A309654 6969696,7777777,8686868,8866688,8884888,9669669,9993999,18181818,26666664,36484836
%N A309654 Numbers k such that k multiplied by the sum of reciprocals of digits is the digit reversal of k.
%C A309654 Integers with at least one 0 digit cannot be terms. There is no other palindrome after 999999999, but is the sequence complete? The non-palindromic numbers in the sequence are 864, 362436, 18181818, 26666664, 36484836, 48363648 and 1666516665, in which zero, seven and nine do not appear as a digit. These non-palindromes are multiples of 3 and have a multiple of 6 as the sum of their digits.
%C A309654 The sequence is finite because the sum of the reciprocals of the digits of every zeroless number greater than 10^81-1 exceeds 9, while the ratio R(n)/n is always smaller than 9. a(43) > 10^13, if it exists. - _Giovanni Resta_, Aug 12 2019
%H A309654 Stéphane Rézel, <a href="/A309654/b309654.txt">Table of n, a(n) for n = 1..42</a>
%e A309654 864 is in the sequence because 864 * (1/8 + 1/6 + 1/4) = 468, the digit reversal of 864.
%t A309654 Select[Range[365*10^5],#*Total[1/IntegerDigits[#]]==IntegerReverse[#]&]//Quiet (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 09 2020 *)
%o A309654 (PARI) isok(k) = my(d=digits(k)); if (vecmin(d), k*sum(i=1, #d, 1/d[i]) == fromdigits(Vecrev(d))); \\ _Michel Marcus_, Aug 11 2019
%o A309654 (Magma) [k:k in [1..4000000]| not 0 in Set(Intseq(k))  and k*(&+[1/Intseq(k)[i]:i in [1..#Intseq(k)]]) eq Seqint(Reverse(Intseq(k)))]; // _Marius A. Burtea_, Aug 11 2019
%Y A309654 Cf. A037268.
%K A309654 nonn,base,fini
%O A309654 1,2
%A A309654 _Stéphane Rézel_, Aug 11 2019