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

A388038: Contribution to the OEIS (a(n) is the number of steps to reach 1 when the Sisyphus sequence (A350877) begins with n, or -1 if 1 is never reached.)

 A388038

a(n) is the number of steps to reach 1 when the Sisyphus sequence (A350877) begins with n, or -1 if 1 is never reached.

0

0, 1, 5, 2, 979, 6, 8, 3, 13, 980, 6, 7, 209, 9, 980, 4, 8, 14, 9, 981, 10, 7, 14, 8, 980, 210, 7, 10, 20, 981, 210, 5, 14, 9, 981, 15, 13, 10, 9, 982, 210, 11, 10, 8, 806, 15, 11, 9, 9, 981, 15, 211, 980, 8, 981, 11, 11, 21, 8, 982, 16, 211, 21, 6, 981, 15, 211

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

OFFSET

1,3

COMMENTS

Rules of the Sisyphus sequence (A350877): if the last term, s(m), is even, s(m+1) = s(m)/2; otherwise, s(m+1) = s(m) + the smallest prime not yet added.

Beginning the Sisyphus sequence with the seed 1008973 runs more than 2^30 terms without reaching 1 (A388141). Michael S. Branicky found that A388141(3584971546) = 1. - Michael De Vlieger, Sep 14 2025

LINKS

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

Michael De Vlieger, Scatterplot of first 65536 terms.

Michael De Vlieger, The Sisyphus Sequence, 2022.

EXAMPLE

For n = 1, a(n) = 0, no steps are required to reach 1.

For n = 3, a(n) = 5.

Step 1: 3 is odd, the next term is 3 + 2 (first prime) = 5.

Step 2: 5 is odd, the next term is 5 + 3 (next prime) = 8.

Step 3: 8 is even, the next term is 8/2 = 4.

Step 4: 4 is even, the next term is 4/2 = 2.

Step 5: 2 is even, the next term is 2/2 = 1.

MATHEMATICA

a[n_]:=Module[{s={n}, p=2, c=1}, While[s[[-1]]!=1, If[OddQ[s[[-1]]], AppendTo[s, s[[-1]]+p]; p=NextPrime[p], AppendTo[s, s[[-1]]/2]]; c++]; c-1]; Array[a, 67]

PROG

(Python)

from sympy import nextprime

def sisyphus(start): # generator of Sisyphus sequence beginning at start

an, p = start, 1

while True: yield an; an = an+(p:=nextprime(p)) if an&1 else an>>1

def a(n): return next(k for k, t in enumerate(sisyphus(n)) if t == 1)

print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Sep 16 2025

CROSSREFS

Cf. A350877, A388141.

Sequence in context: A347426 A212481 A046270 * A139207 A137237 A369841

Adjacent sequences: A388035 A388036 A388037 * A388039 A388040 A388045

KEYWORD

look,sign,new

AUTHOR

James C. McMahon, Sep 13 2025

STATUS

approved

Important Links: https://vincico.com/seq/a350877.html

Scatterplot of terms:

Why do terms cluster near certain values (see horizontal bands in below diagram):

Also see: https://jamesmacmath.blogspot.com/2025/09/the-sisyphus-sequence-a350877.html.

The Sisyphus Sequence (A350877)

(Image: Sisyphus (1548–49) by Titian, Prado Museum, Madrid, Spain)

Sisyphus, in Greek mythology, was cursed upon death to roll a large boulder up a hill only for it to roll down once it neared the top, repeating for eternity. 

The integer sequence, OEIS:  (https://oeis.org/A350877) starts 1, 3, 6, 3, 8, 4, 2, 1, 8 ,4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1... . The formal definition for the sequence is: start the sequence S with a(1) = 1 and extend S with a(n)/2 when a(n) is even, otherwise with a(n) + the smallest prime not yet added. The sequence was named Sisyphus because the sequence climbs only to drop back to 1.

Additional notes on the sequence:  Michael De Vlieger's OEIS sequence analysis: A350877.

A proposed variant of this sequence is to begin the sequence with n, integers 1 and greater, and define a(n) as the number of steps required to reach 1. This new sequence is: 

0, 1, 5, 2, 979, 6, 8, 3, 13, 980, 6, 7, 209, 9, 980, 4, 8, 14, 9, 981, 10, 7, 14, 8, 980, 210, 7, 10, 20, 981, 210, 5, 14, 9, 981, 15, 13, 10, 9, 982, 210, 11, 10, 8, 806, 15, 11, 9, 9, 981, 15, 211, 980, 8, 981, 11, 11, 21, 8, 982, 16, 211, 21, 6, 981, 15, 211...

a(1) = 0 because no further steps are required to reach 1.

a(2) = 1 because in one step, 2/2 =1, 1 is reached.

a(3) = 5 the steps proceed: 5, 8, 4, 2, 1 (5 steps to reach 1).

An interesting feature of this new sequence is how terms tend to cluster around certain values - from just the first 67 terms above, see terms clustered around 11, 210, 980.

This clustering feature is shown below in the scatterplot of the first 4000 terms:

Michael De Vlieger produced a log-log scatterplot of the first 65536 terms, which shows how this clustering continues:

Sequence published: https://jamesmacmath.blogspot.com/2025/09/a388038-contribution-to-oeis-is-number.html.