A389240 - A contribution to the On-Line Encyclopedia of Integer Sequences (OEIS)

This new sequence was just published today. I was happy to collaborate with my friend and fellow Illinois Tech alumnus and swimmer, Michael De Vlieger.

A389240

Start the sequence S with S(1) = n and extend S with S(m+1) = S(m)/2 when S(m) is even, otherwise with S(m) + the smallest odd number not yet added. a(n) is the number of steps to reach 1, or -1 if 1 is never reached.

0

0, 1, 3, 2, 8, 4, 4, 3, 6, 9, 9, 5, 33, 5, 5, 4, 9, 7, 7, 10, -1, 10, 10, 6, 7, 34, 34, 6, 13, 6, 6, 5, 34, 10, 10, 8, 9, 8, 8, 11, 10, -1, -1, 11, 11, 11, 11, 7, -1, 8, 8, 35, 34, 35, 35, 7, 8, 14, 14, 7, 96, 7, 7, 6, 14, 35, 35, 11, -1, 11, 11, 9, 35, 10, 10

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,3

COMMENTS

For n = 1, S begins 1, 2, 1, 4, 2. The subsequent terms are A066070.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Scatterplot of first 1000000 terms

FORMULA

a(2^k) = k.

a(2^k-1) = k+1 for k > 1.

a(2^k-2) = k+1 for k > 2.

For odd m and a(m) = h, a(2^k*m) = h+k.

EXAMPLE

For n = 5, S begins 5, 6, 3, 6, 3, 8, 4, 2, 1, thus 8 steps to reach 1, so a(5) = 8.

For n = 21, S begins 21, 22, 11, 14, 7, 12, 6, 3, 10, 5, 14, 7, 18, 9. Starting with the 7th step, 3, alternating terms of S are the odd numbers 3, 5, 7, 9..., so the sequence never reaches 1; a(21) = -1.

MATHEMATICA

Table[m = -1;

If[#[[-1]] == 1, -1 + Length[#], -1] &@

If[n < 5,

NestWhileList[If[EvenQ[#], #/2, # + (m += 2)] &, n, # > 1 &],

NestWhileList[If[EvenQ[#], #/2, # + (m += 2)] &, n,

And[#4 > 1, Nand[#1 == #3 - 2, #2 == #4 - 4]] &, 4]], {n, 75}]

CROSSREFS

Cf. A066070, A386819, A389391.

Sequence in context: A019666 A110938 A135852 * A191731 A143515 A082333

Adjacent sequences: A389235 A389236 A389237 * A389241 A389242 A389244

KEYWORD

sign,new

AUTHOR

James C. McMahon and Michael De Vlieger, Sep 26 2025

STATUS

approved

A Rock of Sisyphus Reached One

 

An interesting variation of the Sisyphus Sequence, A350877,  is starting the sequence with 1008973 instead of 1. With most first terms, the sequence eventually reaches 1; however, in the case of 1008973, it didn't appear that 1 would be reached. Then, after 3584971546 terms, 1 is reached. 

This is documented in the Online Encyclopedia of Integer Sequences (OEIS) with sequence https://oeis.org/A388141.

The number of steps required to reach 1 when starting the Sisyphus Sequence with integer n, is A388038.

Other posts:

Sisyphus Sequence.

Steps to reach 1 when starting Sisyphus Sequence with n.

Pythagorean Day

Today's date, 9/16/2025, is Pythagorean Day because 9 - 16 - 25 are the squares of the sides of the 3 - 4 - 5 right-angle triangle. This year is particularly special because 2025 is also the square of 45.

Read more at: https://www.npr.org/2025/09/16/nx-s1-5535545/pythagorean-triple-square-day-9-16-25