WON plate46 | World!OfNumbers [ December 28, 1999 ] More palindromic_free numbers This plate is a sequel to WONplate 28 In a attempt to find the largest possible 'free of palindromic substrings' - or just palfree - numbers - except of course the single digits themselves - like for instance in the following powers : 2 54 = 18014398509481984 - 17 digits 3 67 = 92709463147897837085761925410587 - 32 digits 355 15 = 179236021709762370418314530975341796875 - 39 digits (You've other record numbers of this kind ! Please submit them to me and I'll display them here also.) Carlos B. Rivera F. sent a method to produce infinite large 'free of palindromic substrings' numbers. First he gives two examples To produce the palfree number 123123123123 multiply 123 with repunit 111111111111 and divide by 111 To produce the palfree number 1234123412341234 multiply 1234 with repunit 111111111111 and divide by 1111 and then he provides the General Formula To produce N-N-N-N multiply N*R(k*n)/R(n) k = times N appears n = digits of N R(n) = (10^n–1)/(10–1) = 11....11 (n times) Is this the beginning or the end of the palfree numbers story ? [ December 5, 2021 ] So many years later I resumed the topic and searched for palfree Fibonacci numbers. Here is the largest Fibonacci numbers I could come up with Fibonacci(61) = 2504730781961 - 13 digits The search went all the way up to Fibonacci(1200) so it looks that the above Fibonacci number will be the largest palfree one. The provisionally complete sequence is : 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 15, 16, 17, 18, 21, 23, 25, 27, 30, 32, 33, 37, 39, 42, 48, 58, 61 Strings of zero's are not allowed, otherwise 109 and 130 would be valid as well. Let me redo the exercice but now with factorials or ' n! ' . The sequence (n)! starts like this : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... Alas from factorial 10 (3628800) onwards you see a string of zero's appearing that grows larger and larger. So from here on we will search for palfree factorials with disregard of these (midway or ending) zero's. Let us see if we can extend our sequence with that one restriction. The next one factorial 12 (479001600) is candidate with 2 x 'two consecutive zero's' in its decimal expansion. But then I went all the way up to factorial 100 and gave up. The factorials grow very fast in length and the probability that a palfree factorial (even with the zero exclusion) pops up is very low. So the story ends here with the final provisional sequence 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12 A000046 Prime Curios! Prime Puzzle Wikipedia 46 Le Nombre 46 Numberland 46
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