an elegant algorithm related to Ulam's spiral

[Don Shepherd]

Ulam's spiral is a simple structure with a couple of fascinating characteristics.

Code:


64 63 62 61 60 59 58 57
37 36 35 34 33 32 31 56
38 17 16 15 14 13 30 55
39 18 05 04 03 12 29 54
40 19 06 01 02 11 28 53
41 20 07 08 09 10 27 52
42 21 22 23 24 25 26 51
43 44 45 46 47 48 49 50

The squares of odd numbers (1, 9, 25, 49) follow a diagonal path to the lower right; squares of even numbers (4, 16, 36, 64) follow a similar pattern to the upper left. There are many diagonal paths that include prime numbers. These are easily visible when you extend the spiral to higher values.

There is a very elegant algorithm for determining the non-trivial neighbors of a selected number that was shown to me by member Allen Thomson several years ago. For example, the non-trivial neighbors of number 22 are 7 and 45 (21 and 23 would be trivial neighbors of 22).

I have implemented this algorithm on the HP-12c, HP-17b solver, and the HP-65.

Here is the 17b solver solution. Enter a value for N and solve for ANS1 and ANS2.

Code:


ULAM:0xL(A:2xIP(SQRT(4xN-2)))+
IF(S(ANS1):IF(2xIP(SQRT(4xN-6))<G(A):
N+G(A)+3:N-G(A)+3)-ANS1:
N+G(A)+5-ANS2)

Here is the solution for the 12c. Enter a value and press R/S. One solution will be displayed. Press X<-->Y for the second solution.

Code:


01 STO 2
02 4
03 X
04 2
05 -
06 SQRT
07 INTG
08 2
09 X
10 STO 1
11 RCL 2
12 +
13 5
14 +
15 STO 3
16 RCL 1
17 RCL 2
18 4
19 X
20 6
21 -
22 SQRT
23 INTG
24 2
25 X
26 X<=Y
27 GOTO 35
28 RCL 2
29 RCL 1
30 -
31 3
32 +
33 RCL 3
34 GOTO 00
35 -
36 X=0
37 GOTO 28
38 RCL 2
39 RCL 1
40 +
41 GOTO 31

edited on 3/9/2019 to add HP-65 solution

Here is the solution for the HP-65. Enter the value you want to check and press A. Then press B and C for the two answers.

Code:


01 LBL A
03 STO 1
04 4
05 x
06 2
07 -
08 SQRT
10 INT
12 2
13 x
14 STO 2
15 RTN
16 LBL B
18 RCL 1
19 RCL 2
20 +
21 5
22 +
23 RTN
24 LBL C
26 RCL 1
27 4
28 x
29 6
30 -
31 SQRT
33 INT
35 2
36 x
37 RCL 2
38 X>Y
39 GOTO 1
41 RCL 1
42 RCL 2
43 -
44 3
45 +
46 RTN
47 LBL 1
49 RCL 1
50 RCL 2
51 +
52 3
53 +
54 RTN