A115026 Limiting value of n under iteration of "sum of the digits raised to the power of the number of digits of n" (A101337).

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 1, 370, 370, 370, 370, 370, 1, 4, 5, 8, 1, 4, 370, 370, 370, 1, 370, 9, 1, 1, 370, 370, 370, 370, 370, 370, 370, 370, 370, 4, 370, 1, 370, 370, 370, 370, 1, 370, 370, 370, 370, 370, 370, 370, 370, 370, 160, 370, 370, 370, 370, 370, 370

Offset

1,2

Comments

Iterate A101337 starting at n until reaching a constant value (like 370) or a cycle (like 160, 217, 352, 160, ...). In the latter case, a(n) takes the smallest value in the cycle (e.g., a(59) = 160). Since k*9^k < 10^k for all k > 34, each number n is guaranteed to yield a smaller number a(n) if n > 10^34, so every number reaches a constant or a cycle under this sequence.

Conjecture: no term is greater than 370. - Harvey P. Dale, Jun 08 2022

Links

Table of n, a(n) for n=1..65.

Example

a(89)=370 since:
89 (2 digits): 8^2 + 9^2 = 145,
145 (3 digits): 1^3 + 4^3 + 5^3 = 190,
190 (3 digits): 1^3 + 9^3 + 0^3 = 730,
730 (3 digits): 7^3 + 3^3 + 0^3 = 370,
370 (3 digits): 3^3 + 7^3 + 0^3 = 370, etc.
So a(89) = 370 since 370 is a fixed point of A101337.

Mathematica

Table[Min[FindTransientRepeat[NestList[Total[IntegerDigits[#]^IntegerLength[#]]&, n, 20], 3][[2]]], {n, 70}] (* Harvey P. Dale, Jun 08 2022 *)

Crossrefs

Cf. A101337.

Sequence in context: A355370 A326344 A340270 * A360075 A101337 A135208

Adjacent sequences: A115023 A115024 A115025 * A115027 A115028 A115029

Keyword

nonn,base

Author

Sergio Pimentel, Feb 24 2006

Status

approved