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

Author

Pierre CAMI, Mar 02 2018

Keywords

• nonn

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).

Examples

			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.
		

Links

• Pierre CAMI, PFGW Script

Crossrefs

Cf. A006577, A294748, A297307.

Programs

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