A276202 Smallest integer k such that the number of semiprimes in {k, f(k), f(f(k)), ..., 1} is equal to n, where f is the Collatz function.

1, 4, 3, 6, 11, 7, 14, 9, 25, 33, 59, 57, 39, 135, 191, 127, 123, 159, 219, 379, 251, 167, 223, 111, 793, 263, 175, 466, 103, 137, 183, 91, 121, 107, 71, 47, 31, 41, 27, 97, 145, 129, 171, 231, 235, 553, 411, 487, 327, 649, 703, 763, 1519, 1215, 1471, 1071

Offset: 0

Author

Michel Lagneau, Aug 29 2016

Keywords

• nonn

Comments

Smallest k such that A275866(k) is equal to n.

Conjecture: a(n) exists for any n.

Examples

			a(5)=7 because the trajectory of 7 is 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 and the 5 semiprimes of this trajectory are 22, 34, 26, 10 and 4.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 0..100
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006577, A275866.

Programs

Mathematica

f[n_]:=n/2/; Mod[n, 2]==0; f[n_]:=3 n+1/; Mod[n, 2]==1; g[n_]:=Module[{i, p}, i=n; p=0; While[i>1, If[PrimeOmega[i]==2, p=p+1]; i=f[i]]; p]; Do[k=1; While[g[k]!=m, k++]; Print[m, " ", k], {m, 0, 53}]
nsp[n_]:={n,Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],?(PrimeOmega[#]==2&)]}; Table[SelectFirst[nsp/@Range[2000],#[[2]] == n&],{n,0,59}][[;;,1]] (* _Harvey P. Dale, Mar 29 2025 *)