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

1,2

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.

Links

Table of n, a(n) for n=1..58.

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

Cf. A257588 (iteration step).

Cf. A007770 (sum of squares not alternating).

Sequence in context: A259260 A081352 A190673 * A175651 A372045 A362979

Adjacent sequences: A345677 A345678 A345679 * A345681 A345682 A345683

Keyword

nonn,base

Author

Stephen Cross, Jun 23 2021

Status

approved