A300286 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.

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

Offset

0,1

Comments

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).

Links

Table of n, a(n) for n=0..36.

Pierre CAMI, PFGW Script

Example

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.

Prog

(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

Crossrefs

Cf. A006577, A294748, A297307.

Sequence in context: A193151 A034209 A260614 * A058430 A079286 A275685

Adjacent sequences: A300283 A300284 A300285 * A300287 A300288 A300289

Keyword

nonn

Author

Pierre CAMI, Mar 02 2018

Status

approved