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

1, 22, 333, 424, 864, 3663, 4444, 6336, 39993, 46664, 48484, 55555, 64646, 66366, 84448, 88288, 93939, 362436, 488884, 666666, 848848, 884488, 6699966, 6886886, 6969696, 7777777, 8686868, 8866688, 8884888, 9669669, 9993999, 18181818, 26666664, 36484836

Offset: 1

Author

Stéphane Rézel, Aug 11 2019

Keywords

• nonn
• base
• fini

Comments

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.

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

Examples

			864 is in the sequence because 864 * (1/8 + 1/6 + 1/4) = 468, the digit reversal of 864.
		

Links

• Stéphane Rézel, Table of n, a(n) for n = 1..42

Crossrefs

Cf. A037268.

Programs

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

Mathematica

Select[Range[365*10^5],#*Total[1/IntegerDigits[#]]==IntegerReverse[#]&]//Quiet (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 09 2020 *)

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