A275866 -id:A275866 - cp's OEIS frontend

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

Michel Lagneau, Aug 29 2016

nonn

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

Conjecture: a(n) exists for any n.

			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.

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

Cf. A006577, A275866.

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 *)

A277336 Numbers n for which the sum of the odd members and the sum of the even members in the Collatz (3x+1) trajectory are both semiprime.

Original entry on oeis.org

6, 12, 24, 35, 61, 76, 96, 118, 146, 162, 230, 245, 338, 362, 384, 426, 444, 460, 472, 580, 584, 605, 642, 645, 664, 697, 718, 740, 790, 804, 812, 814, 830, 852, 877, 920, 926, 954, 979, 1098, 1178, 1192, 1216, 1332, 1334, 1406, 1415, 1446, 1452, 1454, 1459

Offset: 1

Michel Lagneau, Oct 09 2016

nonn

The corresponding pairs of semiprimes are (9, 46), (9, 58), (9, 82), (94, 446), (178, 838), (95, 538), (9, 226), (411, 1894), (20499, 82366), (259, 1366), (493, 2446), (362, 1942), ...

			6 is in the sequence because the Collatz trajectory is 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 => the sum of the odd members is 3 + 5 + 1 = 9 = 3*3 and the sum of the even members is 6 + 10 + 16 + 8 + 4 + 2 = 46 = 2*23.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A213909, A213916, A275866.

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&];a:=Select[coll[n],OddQ[#]&];b:=Select[coll[n],EvenQ[#]&];Do[s1=Sum[a[[i]],{i,1,Length[a]}];s2=Sum[b[[j]],{j,1,Length[b]}];If[PrimeOmega[s1]==2&&PrimeOmega[s2]==2,Print[n]],{n,1,1500}]

is(n)=my(e,o=1); while(n>1, if(n%2, o+=n; n+=2*n+1, e+=n; n/=2)); isprime(e/2) && bigomega(o)==2 \\ Charles R Greathouse IV, Oct 09 2016

cp's OEIS Frontend

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.

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