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.