The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A119733 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A119733

Offsets of the terms of the nodes of the reverse Collatz function.
(history; published version)

#64 by N. J. A. Sloane at Fri Oct 22 23:25:31 EDT 2021

STATUS

proposed

approved

#63 by Wesley Ivan Hurt at Fri Oct 22 21:20:32 EDT 2021

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editing

proposed

#62 by Wesley Ivan Hurt at Fri Oct 22 21:20:26 EDT 2021

FORMULA

a(0) = 0, a(2n 2*n + 1) = 2a2*a(n) + 3^wt(n) = 2a2*a(n) + A048883(n), a(2n2*n) = 2a2*a(n), where wt(n) = A000120(n) = the number 1's in the binary representation of n.

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proposed

editing

#61 by Kevin Ryde at Fri Oct 22 18:35:52 EDT 2021

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editing

proposed

#60 by Kevin Ryde at Fri Oct 22 16:26:31 EDT 2021

LINKS

Víctor Martín Chabrera, <a href="https://upcommons.upc.edu/handle/2117/133344">An algebraic fractal approach to Collatz Conjecture</a>, Bachelor tesis, Universitat Politècnica de Catalunya (Barcelona, 2019), see lemma 6.1 with a(n) = r(A005836(n)).

FORMULA

a(n) = Sum_{i=0..k} 2^e[i] * 3^i where binary expansion n = 2^e[0] + 2^e[1] + ... + 2^e[k] with descending e[0] > e[1] > ... > e[k] (A272011). [Martín Chabrera lemma 6.1, adapting index i] - Kevin Ryde, Oct 22 2021

PROG

(PARI) a(n) = my(ret=0); if(n, for(i=0, logint(n, 2), if(bittest(n, i), ret=3*ret+1<<i))); ret; \\ Kevin Ryde, Oct 22 2021

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approved

editing

#59 by Joerg Arndt at Wed Apr 21 11:25:40 EDT 2021

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reviewed

approved

#58 by Michel Marcus at Wed Apr 21 07:42:15 EDT 2021

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proposed

reviewed

#57 by F. Chapoton at Wed Apr 21 07:39:39 EDT 2021

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editing

proposed

#56 by F. Chapoton at Wed Apr 21 07:39:31 EDT 2021

PROG

def a(n): return 0 if n==0 else 2*a((n - 1)//2) + 3**bin((n - 1)//2).count('1') if n%2==1 else 2*a(n//2)

print map([a, (n) for n in range(131)]) # Indranil Ghosh, Aug 13 2017

STATUS

approved

editing

Discussion

Wed Apr 21

07:39

F. Chapoton: adapt py code to py3

#55 by Jon E. Schoenfield at Tue Dec 17 05:41:32 EST 2019

STATUS

proposed

approved