This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(3) = 14 since A073142(2) = 11436678 is the smallest solution of x = f^14(x).
In base 3, the RATS mapping acts as 1 -> 2 -> 4 (11 in base 3) -> 8 (22 in base 3) -> 13 (112 in base 3) -> 4, which has already been seen 3 steps ago, so a(3)=3.
a(8) = 1017, since 1017 is the smallest number whose 'Reverse and Subtract' trajectory has eight preperiodic terms: 1017 -> 6084 -> 1278 -> 7443 -> 3996 -> 2997 -> 4995 -> 999.
a(0) = -1, since 0 -> |0 - 0| = 0, the preperiodic part is empty; a(12) = 9, since 12 -> |12 - 21| = 9.
6 is a record, since the preperiodic part of the trajectory of 13 has length 6 and for k < 13 the preperiodic part has a smaller length (at most 2).
f:=n->add(floor( n*(n-1)/(2*k) ),k=1..n );
10001145 -> 44108856 -> 21771288 -> 66446424 -> 23981958 and 23981958 as a term of A072142 is the first term of the periodic part of the trajectory of 10001145, period length is 14.
100000114412 -> 114410885589 -> 871177128822 -> 642355357644 -> 195601804398 -> 697806302193 -> 306602693397 -> 486793513206 -> 115521884478 and 115521884478 as a term of A072143 is the first term of the periodic part of the trajectory of 100000114412, period is 22.
a(119) = 4 since |119 - 191| = 72 -> |72 - 27| = 45 -> |45 - 54| = 9 -> |9 - 9| = 0. The value a(0) maps to 0, so the sequence ends there after 4 values have been traversed. a(12737) = 1 since |12737 - 27371| = 14634 -> |14634 - 46341| = 31707 -> |31707 - 17073| = 14634. Since 14634 is already in the sequence, the sequence ends there.
Array[Function[w, LengthWhile[w, # != Last[w] &]]@ NestWhileList[Abs[# - FromDigits@ RotateLeft@ IntegerDigits[#]] &, #, Unequal, All] &, 105, 0] (* Michael De Vlieger, Apr 13 2021 *)
def roll(n):
""" Moves first digit to the back """
s = str(n)
return int(s[1:] + s[0])
def backtrack(past, length, offset, dct):
""" Goes through every value passed and adds it and it's length to the dictionary """
if length == 0:
for elem in past:
dct[elem] = 0
i = 0
while length > 0:
n = past[i]
dct[n] = length + offset
i += 1
length -= 1
return dct
def a(n, dct):
past = []
length = 0
while (n not in dct):
past.append(n)
length += 1
n = abs(n - roll(n))
if n in past: # For duplicates
length = past.index(n)
dct = backtrack(past, length, 0, dct)
return dct, length
offset = dct[n]
dct = backtrack(past, length, offset, dct)
length += offset
return dct, length
dct = {}
sequence = []
i = 1
while i < 1000:
out = a(i, dct)
dct = out[0]
sequence.append(out[1])
i += 1
f(16) = 61-16 = 45, f(45) = 54-45 = 9, f(9) = 9-9 = 0. For no k < 16 exactly three steps lead to 0, hence a(3) = 16.
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