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

1,1

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

Links

Table of n, a(n) for n=1..62.

Example

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 -> ... .

Mathematica

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

Prog

(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

Crossrefs

Subsequence of A344208.

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

Sequence in context: A314003 A314004 A296031 * A314005 A164355 A287322

Adjacent sequences: A344211 A344212 A344213 * A344215 A344216 A344217

Keyword

nonn,base

Author

Joseph Brown, May 11 2021

Status

approved