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A300286
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Define a set of generalized Syracuse sequences starting with x(1)=2*n+1 a positive odd integer, if x(i) is odd prime set x(i+1)=67*x(i)+1, if x(i) is odd not prime set x(i+1)=3*x(i)+1 and if x(i) is even then set x(i+1)=x(i)/2. Then a(n) is the first index i > 1 at which x(i) reaches 1.
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3
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4, 209166, 13, 207226, 207229, 384614, 384602, 32, 104820, 403030, 8, 30, 403033, 118516, 39365, 403070, 403036, 118323, 11641, 118425, 118514, 89369, 104824, 180241, 11644, 39371, 118321, 118294, 89372, 118423, 119595, 39372, 11647, 403093, 384607, 47436, 124886
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OFFSET
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0,1
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COMMENTS
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I define the generalized Syracuse sequences as follows:
Start with an odd positive number x(1)=2*k+1; then, for i >= 1, if x(i) is an odd prime set x(i+1)=p*x(i)+1 with p a prime, if x(i) is an odd nonprime set x(i+1)=3*x(i)+1, and if x(i) is even then set x(i+1)=x(i)/2.
If p=3 the sequences are the Syracuse sequences in which it does not matter whether odd x(i) is prime or not.
For all the prime numbers p other than 3, if x(i) is odd, the value of x(i+1) depends on whether x(i) is prime.
Among prime numbers p < 97, 67 is the only one for which x(i) reaches 1 for any k < 125 and for k=125, x(1)=251, x(8113)=887, x(8113+8099)=887 a cycle of 8099 values.
All the sequences for p < 423 eventually enter a loop (not tested above, but I conjecture that it is the case for any prime, although with different end cycles).
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LINKS
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Table of n, a(n) for n=0..36.
Pierre CAMI, PFGW Script
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EXAMPLE
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For p=67 and k=2, we have x(1)=2*2+1=5, x(2)=67*5+1=336, x(3)=336/2=168, x(4)=168/2=84, x(5)=84/2=42, x(6)=42/2=21, x(7)=3*21+1=64, x(8)=64/2=32, x(9)=32/2=16, x(10)=16/2=8, x(11)=8/2=4, x(12)=4/2=2, x(13)=2/2=1; x(i) reaches 1 at i=13, so a(2)=13.
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PROG
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(PARI) f(n) = if (n % 2, if (isprime(n), 67*n+1, 3*n+1), n/2);
a(n) = {my(k = f(2*n+1), nb = 2); while (k != 1, k = f(k); nb++); nb; } \\ Michel Marcus, Mar 28 2018
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CROSSREFS
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Cf. A006577, A294748, A297307.
Sequence in context: A193151 A034209 A260614 * A058430 A079286 A275685
Adjacent sequences: A300283 A300284 A300285 * A300287 A300288 A300289
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KEYWORD
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nonn
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AUTHOR
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Pierre CAMI, Mar 02 2018
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STATUS
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approved
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