Hello Math-Fun, is this seq finite (and correct)? S = 1,2,3,11,4,22,101,5,111,6,1001,7,88,77,8,1111,9,... (my knowledges about repunits and palindromes are close to the palindrome 0 themselves) Definition: Lexicographically earliest seq of distinct palindromes such that the product of two successive palindromes is a palindrome. Best, É.
I have convinced myself (just with a calculator) that the next two entries are 10001 and 33. I am pretty sure that the sequence is infinite. 10^n + 1 = 100...01 is always a palindrome, and no matter what the last entry K is, a big enough n will avoid carries to produce a palindrome of the form K0...0K. This trick alone lets us produce an infinite sequence of distinct palindromes whose pairwise products are also palindromes, and the existence of at least one infinite sequence guarantees that there is a lexicographic minimum one. On Sun, Jan 12, 2020 at 1:22 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Hello Math-Fun, is this seq finite (and correct)? S = 1,2,3,11,4,22,101,5,111,6,1001,7,88,77,8,1111,9,... (my knowledges about repunits and palindromes are close to the palindrome 0 themselves) Definition: Lexicographically earliest seq of distinct palindromes such that the product of two successive palindromes is a palindrome. Best, É.
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AW: "10^n + 1 = 100...01 is always a palindrome, and no matter what the last entry K is, a big enough n will avoid carries to produce a palindrome of the form K0...0K." Indeed. More generally, palindromes composed of only the digits zero and one are a great help in advancing the sequence. The fourteen terms at even indices from #166 to #192 are all such: ... 22322, 10111101, 10701, 11100111, 4224, 100010001, 373, 100101001, 383, 101000101, 393, 101010101, 464, 1000000001, 474, 1001001001, 484, 1010000101, 494, 10000000001, 505, 110000011, 515, 110010011, 525, 1000110001, 535, 1100000011, 545, ...
Thanks to AW and HH -- yes, the seq is infinite. The 0 and 1 palindromes will draw their own line in the graph, I guess. Best, É.
Le 13 janv. 2020 à 16:53, Hans Havermann <gladhobo@bell.net> a écrit :
AW: "10^n + 1 = 100...01 is always a palindrome, and no matter what the last entry K is, a big enough n will avoid carries to produce a palindrome of the form K0...0K."
Indeed. More generally, palindromes composed of only the digits zero and one are a great help in advancing the sequence. The fourteen terms at even indices from #166 to #192 are all such: ... 22322, 10111101, 10701, 11100111, 4224, 100010001, 373, 100101001, 383, 101000101, 393, 101010101, 464, 1000000001, 474, 1001001001, 484, 1010000101, 494, 10000000001, 505, 110000011, 515, 110010011, 525, 1000110001, 535, 1100000011, 545, ... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
EA: "S = 1,2,3,11,4,22,101,5,111,6,1001,7,88,77,8,1111,9,..." This is going to be < https://oeis.org/A332661 > once approved. I managed to get 303 terms. #300 is 10^16+1, the largest number to that point. #301-303 are 757, 8421248, and 75700000757. #304 > (10^17-1)/9, which was the largest entry in my palindrome database.
I managed to get 303 terms.
... geee, Hans, this is unbelievable, what a marvel! Can't wait to see the terms -- many thanks for the submission! Best, É. Catapulté de mon aPhone
Le 18 févr. 2020 à 23:19, Hans Havermann <gladhobo@bell.net> a écrit :
EA: "S = 1,2,3,11,4,22,101,5,111,6,1001,7,88,77,8,1111,9,..."
This is going to be < https://oeis.org/A332661 > once approved. I managed to get 303 terms. #300 is 10^16+1, the largest number to that point. #301-303 are 757, 8421248, and 75700000757. #304 > (10^17-1)/9, which was the largest entry in my palindrome database. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Allan Wechsler -
Hans Havermann -
Éric Angelini