A339773 a(n) is the first number k such that the n Collatz runs starting at the consecutive numbers k, k+1, ..., k+n-1 all have the same prime-valued height while the runs starting at k-1 and k+n have nonprime heights.

25, 14, 108, 314, 1154, 840, 3360, 1494, 24408, 4722, 6576, 33578, 124097, 61442, 99248, 104879, 228296, 302956, 203436, 269698, 106122, 470826, 614402, 701224, 589826, 1369884, 252548, 1377184, 3126172, 1356161, 1370050, 1591584, 2065786, 8363804, 2054827

Offset: 1

Author

Hartmut F. W. Hoft, Dec 16 2020

Keywords

• nonn

Comments

The height of a Collatz run starting at a number m is the number of steps to reach 1, A006577(m).

There are three additional blocks with starting values less than 5000000: a(35, 40, 49) = (2054827, 596310, 4330040); a(34) = 8363804. Altogether there are 55 blocks with starting values at most 50000000, the highest of which is a(47) = 37669696 while a(45) > 50000000.

After searching up to about k = 5.3595*10^12, the largest-indexed term observed in the sequence thus far is a(1770) = 2490262807816 which begins a string of 1770 numbers whose Collatz sequence height is 331. - Kevin P. Thompson, Aug 27 2022

After searching up to about k = 3.293*10^13, the largest-indexed term observed in the sequence thus far is a(2225) = 23969528245354 which begins a string of 2225 numbers whose Collatz sequence height is 373. - Kevin P. Thompson, May 28 2023

Examples

			a(2) = 14 since the 2 adjacent numbers 14 and 15 are the first consecutive 2 whose height in their Collatz runs is the same prime number, in this case 17, while the heights for the Collatz runs at 13 and 16 are the nonprimes 9 and 4, respectively.
a(9) = 24408 since the 9 adjacent numbers 24408 .. 24416 are the first consecutive 9 whose height in their Collatz runs is the same prime number, in this case 157, while the heights for the Collatz runs at 24407 and 24417 are the nonprimes 64 and 69, respectively.
		

Links

• Kevin P. Thompson, Table of n, a(n) for n = 1..784
• Kevin P. Thompson, Table of n, a(n) for n = 1..1001 with missing values

Crossrefs

Cf. A006577, A008908, A280341.

Programs

Mathematica

collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
(* b is an estimate on the size of the list being computed *)
a339773[n_, b_] := Module[{k=2, c, d, j, pList=Table[0, {b}]}, While[k<=n, c=height[k-1]; d=height[k]; j=k+1; If[!PrimeQ[c]&&PrimeQ[d], While[height[j]==d, j++]; If[!PrimeQ[height[j]]&&pList[[j-k]]==0, pList[[j-k]]=k]]; k=j]; pList]
Take[a339773[5000000, 50], 33] (* sequence data *)

Extensions

a(34)-a(35) from Kevin P. Thompson, Aug 27 2022