A172143 Numbers of form 4^(3*k+l+1)/9 - 4^l/9 - 1/3 or 2*4^(3*k+l+2)/9 - 2*4^l/9 - 1/3, k,l>=0.

0, 1, 3, 5, 13, 21, 28, 53, 85, 113, 213, 227, 341, 453, 853, 909, 1365, 1813, 1820, 3413, 3637, 5461, 7253, 7281, 13653, 14549, 14563, 21845, 29013, 29125, 54613, 58197, 58253, 87381, 116053, 116501, 116508, 218453, 232789, 233013, 349525, 464213, 466005, 466033

Offset

1,3

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = (A172126(n) - 1)/3. This can be shown by directly deriving the formula in the sequence definition from Collatz rules, by asking, when is (4^{k+1} - 1)/3 * 2^l congruent 1 mod 3? - Ralf Stephan, Dec 18 2025

Mathematica

seq[max_] := Module[{kmax = Floor[Log[4, 3*max+1]], s = {}, s1, odd}, Do[odd = (4^k-1)/3; s1 = 2^Range[0, Floor[Log2[max/odd]]] * odd; s = Join[s, s1], {k, 1, kmax}]; Select[(Union[s] - 1)/3, IntegerQ]]; seq[10^7] (* Amiram Eldar, Sep 01 2024 *)

Prog

(PARI) for(n=1, 2000000, my(o=3*n/2^valuation(n, 2)+1, b=ispower(o)); if(b&&b%2==0&&round(sqrtn(o, b/2))==4&&(n-1)%3==0, print1((n-1)/3, ", ")))

Crossrefs

Cf. A172126.

Sequence in context: A034484 A290441 A222752 * A218790 A168388 A059872

Adjacent sequences: A172140 A172141 A172142 * A172144 A172145 A172146

Keyword

nonn

Author

Ralf Stephan, Nov 19 2010

Extensions

More terms from Amiram Eldar, Sep 01 2024

Definition changed by Ralf Stephan, Dec 18 2025

Status

approved