A341516 The Collatz or 3x+1 function T (A014682) conjugated by unary-binary-encoding (A156552).

1, 3, 2, 6, 3, 7, 5, 12, 4, 27, 7, 14, 11, 75, 6, 24, 13, 35, 17, 54, 10, 147, 19, 28, 9, 363, 8, 150, 23, 13, 29, 48, 14, 507, 15, 70, 31, 867, 22, 108, 37, 343, 41, 294, 12, 1083, 43, 56, 25, 63, 26, 726, 47, 175, 21, 300, 34, 1587, 53, 26, 59, 2523, 20, 96, 33, 847, 61, 1014, 38, 243, 67, 140, 71, 2883, 18, 1734

Offset: 1

Author

Antti Karttunen, Feb 15 2021

Keywords

• nonn

Comments

Collatz-conjecture can be formulated via this sequence by postulating that all iterations of a(n), starting from any n > 1, will eventually end reach the cycle [2, 3].

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to 3x+1 (or Collatz) problem
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A005940, A014682, A064989, A156552, A329603.

Cf. A341515 for a variant.

Programs

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341516(n) = if(n%2, A064989(n), A064989(A329603(n)));

Formula

If n is odd, then a(n) = A064989(n), otherwise a(n) = A064989(A329603(n)).

a(n) = A005940(1+A014682(A156552(n))).