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A334226
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a(n) is the number of times that the number 3*k+1 from the Collatz trajectory of n is greater than n.
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1
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0, 0, 2, 0, 1, 2, 5, 0, 6, 1, 4, 1, 2, 5, 5, 0, 2, 5, 5, 0, 1, 3, 3, 0, 6, 1, 40, 3, 4, 4, 38, 0, 7, 2, 2, 2, 3, 4, 9, 0, 39, 1, 5, 1, 2, 3, 37, 0, 3, 4, 4, 0, 1, 40, 40, 0, 5, 1, 5, 3, 4, 38, 38, 0, 3, 3, 3, 0, 1, 2, 35, 0, 40, 1, 3, 1, 2, 6, 6, 0, 4, 38, 38, 0, 1, 4, 4, 0, 3, 1, 31, 2, 3, 36, 36, 0, 41, 2, 2, 0
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OFFSET
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1,3
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COMMENTS
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The Collatz trajectory of the number n is a sequence starting with n and ending with 1 (obtained for the first time), constructed using even steps k to k/2 and odd steps k to 3*k+1. a(n) is the number of times in the Collatz trajectory of n that the number 3*k+1, obtained after the odd step k to 3*k+1, is greater than n. Alternatively, a(n) represents the number of odd terms in the Collatz trajectory of n that are greater than (n-1)/3. a(1)=0 since the Collatz trajectory of 1 has no steps.
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LINKS
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FORMULA
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EXAMPLE
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The Collatz trajectory of n=3 is 3, (10), 5, (16), 8, 4, 2, 1. It happens twice that the number 3*k+1 in this process is greater than n (those numbers 3*k+1 are in parentheses), so a(3)=2.
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MATHEMATICA
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a[n_] := Block[{c = NestWhileList[ If[ EvenQ@ #, #/2, 3 # + 1] &, n, #>1 & ]}, Length@ Select[ Range[2, Length[c]], OddQ[c[[# - 1]]] && c[[#]] > n &]]; Array[a, 90] (* Giovanni Resta, May 19 2020 *)
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PROG
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(PARI) a(n) = my(res=0, cn=n); while(n>1, if(bitand(n, 1), n=3*n+1; if(n>cn, res++); , n>>=1)); res \\ David A. Corneth, May 20 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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