cp's OEIS Frontend

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.

%I A344214 #47 Jun 12 2021 09:19:53
%S A344214 5,11,15,18,19,20,24,27,28,33,36,37,39,42,45,46,48,51,54,55,57,60,63,
%T A344214 64,66,69,72,73,75,78,81,82,84,87,90,91,93,96,99,101,105,108,109,110,
%U A344214 114,117,118,123,126,127,129,132,135,136,138,141,144,145,147,150,153,154
%N 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.
%C A344214 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
%e A344214 11 is in the list because (1+1)^2 + 1 = 5.
%e A344214 12 is not in the list because repeatedly iterating the function starting with f(1) = 12 will yield 2 before 5.
%e A344214 13 is not in the list because it will never yield 5. Specifically, 13 -> 17 -> 65 -> 122 -> 26 -> 65 -> ... .
%t A344214 Select[Range@100,Last@NestWhileList[Total[IntegerDigits@#]^2+1&,#,#>10&&#!=26&]==5&] (* _Giorgos Kalogeropoulos_, May 12 2021 *)
%o A344214 (Python)
%o A344214 def f(n):
%o A344214     s = 0
%o A344214     while n > 0:
%o A344214         s, n = s+n%10, n//10
%o A344214     return s*s+1
%o A344214 n, pota = 0, 0
%o A344214 while n < 62:
%o A344214     a, repf, i, ii = pota, 0, 0, 4
%o A344214     while a > 9 and a != repf:
%o A344214         a, i = f(a), i+1
%o A344214         if i == ii:
%o A344214             repf, ii = a, 2*ii
%o A344214     if a == 5:
%o A344214         n = n+1
%o A344214         print(pota, end = ", ")
%o A344214     pota = pota+1 # _A.H.M. Smeets_, May 13 2021
%o A344214 (Python)
%o A344214 def f(x): return sum(map(int, str(x)))**2 + 1
%o A344214 def ok(n):
%o A344214   iter = n  # set to f(n) if number of iterations must be >= 1
%o A344214   while iter > 9:
%o A344214     if iter in {65, 122, 26}: return False
%o A344214     iter = f(iter)
%o A344214   return iter == 5
%o A344214 print(list(filter(ok, range(1, 155)))) # _Michael S. Branicky_, May 19 2021
%Y A344214 Subsequence of A344208.
%Y A344214 Cf. A007953, A052216, A052224, A344208, A118881.
%K A344214 nonn,base
%O A344214 1,1
%A A344214 _Joseph Brown_, May 11 2021