This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The Collatz iteration of 7 produces 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and 1, which are either even, prime, or 1.
a177000 n = a177000_list !! (n-1)
a177000_list = filter (all (\x -> even x || a010051' x == 1) .
(init . a070165_row)) a000040_list
-- Reinhard Zumkeller, Apr 03 2012
Collatz[n_] := NestWhileList[If[EvenQ[ # ], #/2, 3#+1] &, n, #>1 &]; Reap[Do[p=Prime[n]; s=Most[Select[Collatz[p],OddQ]]; If[And@@PrimeQ[s], Sow[p]], {n,PrimePi[10^4]}]][[2,1]]
oenQ[n_]:=AllTrue[DeleteCases[Most[NestWhileList[If[EvenQ[#],#/2, 3#+1]&,n, #>1&]], ?PrimeQ],EvenQ]; Select[Prime[Range[5000]], oenQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Dec 28 2014 *)
is(n)=isprime(n) && (n<23 || is((3*n+1)>>valuation(3*n+1,2))) \\ Charles R Greathouse IV, Jun 20 2013
a(1) = 1: the square 1 contained in every trajectory at the end, a(2) = 3: 3 squares in 3 -> 10 -> 5 -> 4^2 -> 8 -> 2^2 -> 2 -> 1^2, a(3) = 9: 4 squares in 3^2 -> 28 -> ... -> 10 -> as above, a(4) = 27: the famous long trajectory A008884 includes the 5 squares 22^2, 11^2, 4^2, 2^2, 1^2, a(5) = 133: 6 squares in 133 -> 20^2 -> 200 -> 10^2 -> 50 -> 5^2 -> ... -> 4^2, 2^2, 1^2, a(6) = 315: 7 squares in 315 -> 946 -> ... -> 533 -> 40^2 -> 800 -> 20^2 -> as above, a(7) = 747: 9 squares in 747 -> 2242 -> 1121 -> 58^2 -> 1682 -> 29^2 -> ... -> 1066 -> 533 -> as above.
nextc(x) = if (x%2==0, x\2, 3*x+1);
a375093(upto=600000) = {my(m=0); for (k=1, upto, np=issquare(k); j=k; while (j>1, j=nextc(j); if (issquare(j), np++)); if (np>m, m=np; print1(k,", ")))}
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