A179738 a(n) = length of (eventual) period of the sequence defined by s(0) = 1, s(n+1) = odd_part(2n-1 + (s(n) if n odd else s(n)*3)), where odd_part = A000265.

1, 2, 4, 1, 4, 8, 2, 4, 8, 4, 2, 2, 1, 4, 12, 4, 4, 26, 2, 12, 6, 2, 4, 4, 4, 16, 4, 8, 10, 6, 4, 8, 4, 2, 6, 8, 12, 12, 28, 1, 22, 16, 2, 16, 12, 12, 16, 6, 4, 40, 24, 4, 32, 6, 26, 16, 6, 8, 4, 8, 12, 2, 36, 6, 46, 12, 2, 36, 60, 6, 16, 8, 4, 18, 20, 4, 60, 36

Offset

2,2

Comments

Original definition (edited): Let x, y be odd numbers and the operation x <+> y := A000265(x+y). Consider sequence s(0) = x <+> y, s(2*k+1) = x <+> 3*s(2*k), s(2*k+2) = x <+> s(2*k+1); a(n) is the smallest period in case x = 2*n-1, y = 1.

The operation x <+> y = A000265(x+y) = odd part of x+y is also considered in A179382.

Record values are: a(2) = 1, a(3) = 2, a(4) = 4, a(7) = 8, a(16) = 12, a(19) = 26, a(40) = 28, a(51) = 40, a(66) = 46, a(70) = 60, a(111) = 64, a(126) = 70, a(147) = 80, a(162) = 96, a(225) = 120, a(379) = 170, a(619) = 184, a(640) = 228, a(727) = 248, a(759) = 256, a(916) = 348, ... - M. F. Hasler, Feb 16 2025

Links

Jason Yuen, Table of n, a(n) for n = 2..10000 (terms 2..1000 from M. F. Hasler)

Example

For n = 4, 2*n-1 = 7, we get: 7 <+> 1 = 1, 7 <+> 3*1 = 5, 7 <+> 5 = 3, 7 <+> 3*3 = 1, and from here on it starts over with 7 <+> 1 = 1, etc., so the period is [1, 5, 3, 1], of length 4, whence a(4) = 4.
For n = 6, 2*n-1 = 11, we get:
  11 <+> 1 = 3, 11 <+> 3*3 =  5, 11 <+>  5  = 1, 11 <+> 3*1 = 7,
  11 <+> 7 = 9, 11 <+> 3*9 = 19, 11 <+> 19 = 15, 11 <+> 3*15 = 7, 11 <+> 7 = 9, ...
Thus we have an eventually periodic sequence with the smallest period 4 (with elements 7, 9, 19, 15). Thus a(6) = 4.

Prog

(PARI) apply( {A179738(n, y=1, T(y, x=2*n-1)=(x+y)>>valuation(x+y, 2))=my(s=[], P); until(, s=concat(s, y=T(3^(#s%2)*y)); for(L=1, #s\3, P=[vecextract(s, Str(-L-t, "..-", 1+t)) | t<-[0, L, 2*L]]; P[1]==P[2] && P[1]==P[3] && return(#P[1])))}, [2..90])
(Python)
def A179738(n):
    s = [1]; x = 2*n-1; odd = lambda z: all(z&1 or(z:=z>>1)for _ in range(z))and z
    while not(p := next((p for p in range(1, len(s)//3+1) if
        s[-p:]==s[-2*p:-p]==s[-3*p:-2*p]), 0)): s.append(odd(x+3**(len(s)&1)*s[-1]))
    return p
print([A179738(n)for n in range(2, 99)]) # M. F. Hasler, Feb 16 2025

Crossrefs

Cf. A179382, A179480, A179686.

Sequence in context: A377126 A201774 A011029 * A256696 A244261 A361873

Adjacent sequences: A179735 A179736 A179737 * A179739 A179740 A179741

Keyword

nonn

Author

Vladimir Shevelev, Jul 25 2010

Extensions

Name edited and corrections proposed by Jason Yuen, Feb 09 2025

Edited, a(4) and a(18) corrected, and extended by M. F. Hasler, Feb 15 2025

Status

approved