OFFSET

1,2

COMMENTS

The sequence was initially studied by a group of students at Clifton College, UK.

There are infinitely many terms.

Having checked up to 10^10, there are approximations for the lower and upper density: 0.23 and 0.25 respectively.

Conjecture: there are strings of consecutive terms of arbitrary length.

Any number which is formed by concatenating two-digit multiples of 11 is a term.

EXAMPLE

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, ...

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 *)

PROG

(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

CROSSREFS

KEYWORD

nonn,base

AUTHOR

Stephen Cross, Jun 23 2021

STATUS

approved