A344214 Numbers k such that repeated iterations of f(m) = (digsum(f(m-1)))^2 + 1 starting from f(1) = k will eventually yield 5 before any other single-digit number.

5, 11, 15, 18, 19, 20, 24, 27, 28, 33, 36, 37, 39, 42, 45, 46, 48, 51, 54, 55, 57, 60, 63, 64, 66, 69, 72, 73, 75, 78, 81, 82, 84, 87, 90, 91, 93, 96, 99, 101, 105, 108, 109, 110, 114, 117, 118, 123, 126, 127, 129, 132, 135, 136, 138, 141, 144, 145, 147, 150, 153, 154

Offset: 1

Author

Joseph Brown, May 11 2021

Keywords

• nonn
• base

Comments

f(x) = digsum(x)^2 + 1 < x for x >= 400, and all iterations terminate in a single digit or lead to the cycle 65 -> 122 -> 26. - Michael S. Branicky, May 14 2021

Examples

			11 is in the list because (1+1)^2 + 1 = 5.
12 is not in the list because repeatedly iterating the function starting with f(1) = 12 will yield 2 before 5.
13 is not in the list because it will never yield 5. Specifically, 13 -> 17 -> 65 -> 122 -> 26 -> 65 -> ... .
		

Crossrefs

Subsequence of A344208.

Cf. A007953, A052216, A052224, A344208, A118881.

Programs

Mathematica

Select[Range@100,Last@NestWhileList[Total[IntegerDigits@#]^2+1&,#,#>10&&#!=26&]==5&] (* Giorgos Kalogeropoulos, May 12 2021 *)

Python

def f(n):
    s = 0
    while n > 0:
        s, n = s+n%10, n//10
    return s*s+1
n, pota = 0, 0
while n < 62:
    a, repf, i, ii = pota, 0, 0, 4
    while a > 9 and a != repf:
        a, i = f(a), i+1
        if i == ii:
            repf, ii = a, 2*ii
    if a == 5:
        n = n+1
        print(pota, end = ", ")
    pota = pota+1 # A.H.M. Smeets, May 13 2021

Python

def f(x): return sum(map(int, str(x)))**2 + 1
def ok(n):
  iter = n  # set to f(n) if number of iterations must be >= 1
  while iter > 9:
    if iter in {65, 122, 26}: return False
    iter = f(iter)
  return iter == 5
print(list(filter(ok, range(1, 155)))) # Michael S. Branicky, May 19 2021