A336992 The number of gaps in the sets A(n), where A(0) is the empty set and A(n+1) is the union of A(n) and the Collatz orbit of the smallest natural number missing in A(n).

0, 0, 1, 3, 3, 9, 9, 8, 12, 12, 14, 15, 15, 16, 100, 99, 100, 100, 101, 108, 108, 112, 111, 110, 110, 110, 110, 111, 110, 116, 115, 115, 115, 116, 120, 120, 124, 123, 122, 122, 121, 122, 122, 123, 125, 124, 125, 125, 126, 125, 125, 127, 127, 126, 133, 133

Offset: 0

Author

Markus Sigg, Aug 10 2020

Keywords

• nonn

Comments

The number of gaps would be relevant for sparse representations of the sets A(n), which may be of use for a numerical verification of the Collatz conjecture up to a given number.

Links

• Markus Sigg, Table of n, a(n) for n = 0..999

Crossrefs

Cf. A061641, A336938.

Programs

PARI

firstMiss(A) = { my(i); if(#A == 0 || A[1] > 0, return(0)); for(i = 1, A[#A] + 1, if(!setsearch(A, i), return(i))); };
iter(A) = { my(a = firstMiss(A)); while(!setsearch(A, a), A = setunion(A, Set([a])); a = if(a % 2, 3*a+1, a/2)); A; };
nGaps(A) = { my(i,c=0); for (i=2, #A, if (A[i-1] < A[i]-1, c = c+1;)); c; };
makeVec(m) = { my(v = [], A = Set([]), i); for(i = 1, m, v = concat(v, nGaps(A)); if (i < m, A = iter(A))); v; };
makeVec(57)