A345680 Nonnegative integers whose trajectory under iteration of taking the absolute value of the alternating sum of the squares of the digits (A257588) includes zero.
0, 7, 11, 22, 29, 33, 34, 38, 43, 44, 47, 49, 55, 56, 59, 65, 66, 70, 74, 77, 83, 88, 92, 94, 95, 99, 108, 110, 117, 125, 126, 131, 138, 142, 147, 148, 149, 161, 168, 171, 172, 179, 182, 184, 185, 195, 196, 205, 212, 220, 227, 234, 237, 238, 241, 258, 265, 269
Offset: 1
Examples
For 7, the trajectory under iteration is 7, 49, 65, 11, 0, ..., so 7 is a term. For 11, the trajectory is 11, 0, ... For 22, the trajectory is 22, 0, ... For 29, the trajectory is 29, 77, 0, ... A non-example is 48. Its trajectory is 48, 48, ...
Programs
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Mathematica
Select[Range[1000], FixedPoint[ Abs[Sum[(-1)^(n + 1)*Part[IntegerDigits[#]^2, n], {n, 1, Length[IntegerDigits[#]]}]] &, #, 10] == 0 &] (* Luca Onnis, Feb 23 2022 *) -
Python
def happyish_function(number, base: int = 10): # A257588 # iterates the process total = 0 times = 0 while number > 0: total += pow(-1, times) * pow(abs(number) % base, 2) number = abs(number) // base times += 1 return abs(total) def is_happyish(number: int) -> bool: # determines whether a number is happyish seen_numbers = set() while number > 0 and number not in seen_numbers: seen_numbers.add(number) number = happyish_function(number) return number == 0 def happyish_list(number: int): # creates their list happyish = [] n = 0 for i in range(number): if is_happyish(i) == True: n +=1 happyish.append(i) return happyish happyish_list(100) # an example
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