Math Vacation

Friday, September 19, 2025

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.

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.

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, 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.