User:Hakerh400/Pair sum
The sequence begins with:
1, 7, 23, 19, 34, 31, 29, 34, 39, 40, 35, 38, 43, 51, 47, 50, 45, 52, 56, ...
How it is formed
Let f be a function that takes a natural number and returns the number formed by the concatenation of the sums of consecutive digits in base 10. For example, consider the number 3571. Calculate the sums 3 + 5 = 8, 5 + 7 = 12, 7 + 1 = 8. Concatenate the sums: 8128. Therefore, f(3571) = 8128. For single-digit numbers, f returns the same number. For example, f(5) = 5.
We say that a natural number n terminates iff there exists k such that fk(n) is a single digit. We say that a number is trivial iff it has at least three digits and all digits except the first and the last are zeros.
Now, consider function fb(n) that behaves just like f, but for any given number base b. This also applies to the definitions of terminating numbers and trivial numbers.
The bth number in the sequence corresponds to the maximal k such that fbk(n) is a single digit for some non-trivial n, and k is the minimal for that n. The sequence starts with b = 2.
Examples
For instance, consider b = 10. The non-trivial number 91000021 reaches digit 4 in 39 steps. It is the maximal number of steps for base 10. Therefore, the corresponding term in the sequence is 39. Here is the step-by-step computation:
91000021 10100023 1110025 221027 43129 74311 11742 28116 10927 19119 1010210 111231 22354 4589 91317 10448 14812 51293 631112 94223 13645 49109 131019 441110 85221 13743 410117 51128 62310 8541 1395 41214 5335 868 1414 555 1010 111 22 4
Another example. For b = 16 the number 0x80000000000000000000000010 reaches digit 0xC in 47 steps:
80000000000000000000000010 8000000000000000000000011 800000000000000000000012 80000000000000000000013 8000000000000000000014 800000000000000000015 80000000000000000016 8000000000000000017 800000000000000018 80000000000000019 800000000000001A 80000000000001B 8000000000001C 800000000001D 80000000001E 8000000001F 80000000110 8000000121 800000133 80000146 800015A 80016F 801715 81886 9910E 12A1E 3CBF F171A 1088B 181013 99114 12A25 3CC7 F1813 10994 1912D AA3F 14D12 511E3 62F11 811102 92212 B433 F76 16D 713 84 C
Conjecture
We conjecture that the sequence has a well-defined value for all b >= 2.