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A275968 Smaller of two consecutive primes p and q such that c(p) = c(q), where c(n) = A008908(n) is the length of x, f(x), f(f(x)), ... , 1 in the Collatz conjecture.

%I A275968 #21 May 22 2025 10:21:44
%S A275968 173,409,419,421,439,487,521,557,571,617,761,887,919,1009,1039,1117,
%T A275968 1153,1171,1217,1327,1373,1549,1559,1571,1657,1693,1709,1721,1733,
%U A275968 1783,1831,1861,1901,1993,1997,2053,2089,2339,2393,2521,2539,2647,2657,2677,2693,2777
%N A275968 Smaller of two consecutive primes p and q such that c(p) = c(q), where c(n) = A008908(n) is the length of x, f(x), f(f(x)), ... , 1 in the Collatz conjecture.
%C A275968 If x is even f(x) = x/2 else f(x) = 3x + 1.
%H A275968 Abhiram R Devesh, <a href="/A275968/b275968.txt">Table of n, a(n) for n = 1..10000</a>
%e A275968 a(1) = p = 173; q = 179
%e A275968 c(p) = c(q) = 32
%t A275968 t = Table[Length@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # != 1 &] - 1, {n, 10^4}]; Prime@ Flatten@ Position[#, k_ /; Length@ k == 1] &@ Map[Union@ Part[t, #] &, #] &@ Partition[#, 2, 1] &@ Prime@ Range@ 410 (* _Michael De Vlieger_, Sep 01 2016 *)
%o A275968 (PARI) A008908(n)=my(c=1); while(n>1, n=if(n%2, 3*n+1, n/2); c++); c
%o A275968 t=A008908(p=2); forprime(q=3,1e4, tt=A008908(q); if(t==tt, print1(p", ")); p=q; t=tt) \\ _Charles R Greathouse IV_, Sep 01 2016
%o A275968 (Python)
%o A275968 import sympy
%o A275968 def lcs(n):
%o A275968     a=1
%o A275968     while n>1:
%o A275968         if n%2==0:
%o A275968             n=n//2
%o A275968         else:
%o A275968             n=(3*n)+1
%o A275968         a=a+1
%o A275968     return(a)
%o A275968 m=2
%o A275968 while m>0:
%o A275968     n=sympy.nextprime(m)
%o A275968     if lcs(m)==lcs(n):
%o A275968         print(m,)
%o A275968     m=n
%o A275968 # _Abhiram R Devesh_, Sep 02 2016
%Y A275968 Cf. A006577 (Collatz trajectory lengths), A078417, A008908.
%K A275968 nonn,easy
%O A275968 1,1
%A A275968 _Abhiram R Devesh_, Aug 15 2016