README.md

Warp Primes

• This repository is the Python 3 code used to generate Warp Primes.
• Warp primes are prime numbers whose reverse of the balanced ternary representation is also prime or -prime.
• Balanced ternary notation might allow insight into relations hidden by classical notation.
• See Wikipedia - Balanced Ternary

Dynamic Ulam Spiral Visualization

• Download and launch the index.html file in a browser to access the D3 visualization.

Warp operator

The warp operator is an unary numerical operator described in A134028: Reversal of n in balanced ternary representation as:

• Convert "classical" number to balanced ternary representation
• Reverse balanced ternary representation
• Convert reversed balanced ternary representation to "classical" number

d3(n) = if ((n%3)==2, n\3+1, n\3);
m3(n) = if ((n%3)==2, -1, n % 3);
t(n) = if (n==0, [0], if (abs(n) == 1, [n], concat(m3(n), t(d3(n)))));
f(n) = subst(Pol(Vec(t(n))), x, 3);
isok(n) = isprime(n) && isprime(abs(f(n))); \\ _Michel Marcus_, Jan 29 2019 

How-to

pip install sympy
py -3 warp.py

Corresponding OEIS sequences

B-files of A323782, A323783 and A323784 are provided for primes < 1000000.

• Warp Entry : OEIS A323782
• Warp Exit : OEIS A323783
• Warp Orphans : OEIS A323784

Related OEIS sequences

• Subsequence of A134028: Reversal of n in balanced ternary representation.
• Supersequence of A???: Prime palindrome in balanced ternary representation.
• To complete...

A323782 & A323783: Prime warpers (primes that warp to other primes)

A323782 Sequence

2,5,7,11,13,17,29,31,37,43,53,59,61,71,73,83,89,101,103,137,139,149,163,173,179,181,193,199,223,233,241,263,269,277,311,313,331,347,353,367,373,379,383,389,401,421,443,449,457,467,479,487,499,509,541,547,569,599,601,607,613,643,677,691,709,719,727,739,751,757,761,773,809,811,823,827,839,853,857,859,863,877,883,887,919,929,947,977,983

A323782 Stats

• Percentage of warp primes below 10: 3 / 3 = 100.0 %
• Percentage of warp primes below 100: 17 / 24 = 70.83333333333334 %
• Percentage of warp primes below 1000: 89 / 167 = 53.293413173652695 %
• Percentage of warp primes below 10000: 474 / 1228 = 38.59934853420196 %
• Percentage of warp primes below 100000: 2870 / 9591 = 29.923886977374618 %
• Percentage of warp primes below 1000000: 18684 / 78497 = 23.80218352293718 %
• Percentage of warp primes below 10000000: 133077 / 664578 = 20.024286088314692 %
• Percentage of warp primes below 100000000: 996574 / 5761454 = 17.297265586082958 %

A323782 & A323783 Table for primes < 500

index	A323782	BT representation	reversed BT	A323783	prime factors
0	2	+ -	- +	-2	[2]
1	5	+ - -	- - +	-11	[11]
2	7	+ - +	+ - +	7	[7]
3	11	+ + -	- + +	-5	[5]
4	13	+ + +	+ + +	13	[13]
5	17	+ - 0 -	- 0 - +	-29	[29]
6	29	+ 0 + -	- + 0 +	-17	[17]
7	31	+ 0 + +	+ + 0 +	37	[37]
8	37	+ + 0 +	+ 0 + +	31	[31]
9	43	+ - - - +	+ - - - +	43	[43]
10	53	+ - 0 0 -	- 0 0 - +	-83	[83]
11	59	+ - + - -	- - + - +	-101	[101]
12	61	+ - + - +	+ - + - +	61	[61]
13	71	+ 0 - 0 -	- 0 - 0 +	-89	[89]
14	73	+ 0 - 0 +	+ 0 - 0 +	73	[73]
15	83	+ 0 0 + -	- + 0 0 +	-53	[53]
16	89	+ 0 + 0 -	- 0 + 0 +	-71	[71]
17	101	+ + - + -	- + - + +	-59	[59]
18	103	+ + - + +	+ + - + +	103	[103]
19	137	+ - - 0 + -	- + 0 - - +	-173	[173]
20	139	+ - - 0 + +	+ + 0 - - +	313	[313]
21	149	+ - 0 - - -	- - - 0 - +	-353	[353]
22	163	+ - 0 0 0 +	+ 0 0 0 - +	241	[241]
23	173	+ - 0 + + -	- + + 0 - +	-137	[137]
24	179	+ - + - 0 -	- 0 - + - +	-263	[263]
25	181	+ - + - 0 +	+ 0 - + - +	223	[223]
26	193	+ - + 0 + +	+ + 0 + - +	331	[331]
27	199	+ - + + 0 +	+ 0 + + - +	277	[277]
28	223	+ 0 - + - +	+ - + - 0 +	181	[181]
29	233	+ 0 0 - 0 -	- 0 - 0 0 +	-269	[269]
30	241	+ 0 0 0 - +	+ - 0 0 0 +	163	[163]
31	263	+ 0 + - + -	- + - + 0 +	-179	[179]
32	269	+ 0 + 0 0 -	- 0 0 + 0 +	-233	[233]
33	277	+ 0 + + - +	+ - + + 0 +	199	[199]
34	311	+ + 0 - - -	- - - 0 + +	-347	[347]
35	313	+ + 0 - - +	+ - - 0 + +	139	[139]
36	331	+ + 0 + - +	+ - + 0 + +	193	[193]
37	347	+ + + 0 - -	- - 0 + + +	-311	[311]
38	353	+ + + 0 + -	- + 0 + + +	-149	[149]
39	367	+ - - - - - +	+ - - - - - +	367	[367]
40	373	+ - - - - + +	+ + - - - - +	853	[853]
41	379	+ - - - 0 0 +	+ 0 0 - - - +	691	[691]
42	383	+ - - - + - -	- - + - - - +	-929	[929]
43	389	+ - - - + + -	- + + - - - +	-443	[443]
44	401	+ - - 0 0 - -	- - 0 0 - - +	-983	[983]
45	421	+ - - + - - +	+ - - + - - +	421	[421]
46	443	+ - - + + + -	- + + + - - +	-389	[389]
47	449	+ - 0 - - 0 -	- 0 - - 0 - +	-839	[839]
48	457	+ - 0 - 0 - +	+ - 0 - 0 - +	457	[457]
49	467	+ - 0 - + 0 -	- 0 + - 0 - +	-677	[677]
50	479	+ - 0 0 - + -	- + - 0 0 - +	-569	[569]
51	487	+ - 0 0 0 0 +	+ 0 0 0 0 - +	727	[727]
52	499	+ - 0 0 + + +	+ + + 0 0 - +	1051	[1051]

A323784: Orphan warp primes (primes that don't warp to primes)

A323784 Sequence

3,19,23,41,47,67,79,97,107,109,113,127,131,151,157,167,191,197,211,227,229,239,251,257,271,281,283,293,307,317,337,349,359,397,409,419,431,433,439,461,463,491,503,521,523,557,563,571,577,587,593,617,619,631,641,647,653,659,661,673,683,701,733,743,769,787,797,821,829,881,907,911,937,941,953,967,971,991,997

A323784 Stats

Stats for A323784 are 100% - stats(A323782) since A323784 is the complement of A323782.

A323784 Table for primes < 500

index	prime	BT representation	reversed BT	warped prime	prime factors
0	3	+ 0	0 +	1	[]
1	19	+ - 0 +	+ 0 - +	25	[5]
2	23	+ 0 - -	- - 0 +	-35	[5, 7]
3	41	+ - - - -	- - - - +	-119	[7, 17]
4	47	+ - - + -	- + - - +	-65	[5, 13]
5	67	+ - + + +	+ + + - +	115	[5, 23]
6	79	+ 0 0 - +	+ - 0 0 +	55	[5, 11]
7	97	+ + - - +	+ - - + +	49	[7]
8	107	+ + 0 0 -	- 0 0 + +	-77	[7, 11]
9	109	+ + 0 0 +	+ 0 0 + +	85	[5, 17]
10	113	+ + + - -	- - + + +	-95	[5, 19]
11	127	+ - - - 0 +	+ 0 - - - +	205	[5, 41]
12	131	+ - - 0 - -	- - 0 - - +	-335	[5, 67]
13	151	+ - 0 - - +	+ - - 0 - +	133	[7, 19]
14	157	+ - 0 - + +	+ + - 0 - +	295	[5, 59]
15	167	+ - 0 + - -	- - + 0 - +	-299	[13, 23]
16	191	+ - + 0 + -	- + 0 + - +	-155	[5, 31]
17	197	+ - + + 0 -	- 0 + + - +	-209	[11, 19]
18	211	+ 0 - - + +	+ + - - 0 +	289	[17]
19	227	+ 0 - + + -	- + + - 0 +	-143	[11, 13]
20	229	+ 0 - + + +	+ + + - 0 +	343	[7]
21	239	+ 0 0 0 - -	- - 0 0 0 +	-323	[17, 19]
22	251	+ 0 0 + 0 -	- 0 + 0 0 +	-215	[5, 43]
23	257	+ 0 + - - -	- - - + 0 +	-341	[11, 31]
24	271	+ 0 + 0 0 +	+ 0 0 + 0 +	253	[11, 23]
25	281	+ 0 + + + -	- + + + 0 +	-125	[5]
26	283	+ 0 + + + +	+ + + + 0 +	361	[19]
27	293	+ + - 0 - -	- - 0 - + +	-329	[7, 47]
28	307	+ + - + 0 +	+ 0 + - + +	265	[5, 53]
29	317	+ + 0 - + -	- + - 0 + +	-185	[5, 37]
30	337	+ + 0 + + +	+ + + 0 + +	355	[5, 71]
31	349	+ + + 0 - +	+ - 0 + + +	175	[5, 7]
32	359	+ + + + 0 -	- 0 + + + +	-203	[7, 29]
33	397	+ - - 0 - 0 +	+ 0 - 0 - - +	637	[7, 13]
34	409	+ - - 0 0 + +	+ + 0 0 - - +	961	[31]
35	419	+ - - + - - -	- - - + - - +	-1037	[17, 61]
36	431	+ - - + 0 0 -	- 0 0 + - - +	-713	[23, 31]
37	433	+ - - + 0 0 +	+ 0 0 + - - +	745	[5, 149]
38	439	+ - - + + - +	+ - + + - - +	583	[11, 53]
39	461	+ - 0 - 0 + -	- + 0 - 0 - +	-515	[5, 103]
40	463	+ - 0 - 0 + +	+ + 0 - 0 - +	943	[23, 41]
41	491	+ - 0 0 + - -	- - + 0 0 - +	-893	[19, 47]
42	503	+ - 0 + - 0 -	- 0 - + 0 - +	-785	[5, 157]
43	521	+ - 0 + + 0 -	- 0 + + 0 - +	-623	[7, 89]
44	523	+ - 0 + + 0 +	+ 0 + + 0 - +	835	[5, 167]
45	557	+ - + 0 - 0 -	- 0 - 0 + - +	-803	[11, 73]
46	563	+ - + 0 0 - -	- - 0 0 + - +	-965	[5, 193]
47	571	+ - + 0 0 + +	+ + 0 0 + - +	979	[11, 89]
48	577	+ - + 0 + 0 +	+ 0 + 0 + - +	817	[19, 43]
49	587	+ - + + - + -	- + - + + - +	-533	[13, 41]
50	593	+ - + + 0 0 -	- 0 0 + + - +	-695	[5, 139]
51	617	+ 0 - - 0 - -	- - 0 - - 0 +	-1007	[19, 53]
52	619	+ 0 - - 0 - +	+ - 0 - - 0 +	451	[11, 41]
53	631	+ 0 - - + 0 +	+ 0 + - - 0 +	775	[5, 31]
54	641	+ 0 - 0 - + -	- + - 0 - 0 +	-575	[5, 23]
55	647	+ 0 - 0 0 0 -	- 0 0 0 - 0 +	-737	[11, 67]
56	653	+ 0 - 0 + - -	- - + 0 - 0 +	-899	[29, 31]
57	659	+ 0 - 0 + + -	- + + 0 - 0 +	-413	[7, 59]
58	661	+ 0 - 0 + + +	+ + + 0 - 0 +	1045	[5, 11, 19]
59	673	+ 0 - + 0 - +	+ - 0 + - 0 +	505	[5, 101]
60	683	+ 0 - + + 0 -	- 0 + + - 0 +	-629	[17, 37]
61	701	+ 0 0 - 0 0 -	- 0 0 - 0 0 +	-755	[5, 151]
62	733	+ 0 0 0 0 + +	+ + 0 0 0 0 +	973	[7, 139]
63	743	+ 0 0 + - - -	- - - + 0 0 +	-1025	[5, 41]
64	769	+ 0 0 + + + +	+ + + + 0 0 +	1081	[23, 47]
65	787	+ 0 + - 0 + +	+ + 0 - + 0 +	955	[5, 191]
66	797	+ 0 + 0 - - -	- - - 0 + 0 +	-1043	[7, 149]
67	821	+ 0 + 0 + + -	- + + 0 + 0 +	-395	[5, 79]
68	829	+ 0 + + - 0 +	+ 0 - + + 0 +	685	[5, 137]
69	881	+ + - 0 - 0 -	- 0 - 0 - + +	-815	[5, 163]
70	907	+ + - + - - +	+ - - + - + +	427	[7, 61]
71	911	+ + - + - + -	- + - + - + +	-545	[5, 109]
72	937	+ + 0 - - 0 +	+ 0 - - 0 + +	625	[5]
73	941	+ + 0 - 0 - -	- - 0 - 0 + +	-995	[5, 199]
74	953	+ + 0 - + 0 -	- 0 + - 0 + +	-671	[11, 61]
75	967	+ + 0 0 - + +	+ + - 0 0 + +	895	[5, 179]
76	971	+ + 0 0 0 0 -	- 0 0 0 0 + +	-725	[5, 29]
77	991	+ + 0 + - 0 +	+ 0 - + 0 + +	679	[7, 97]
78	997	+ + 0 + 0 - +	+ - 0 + 0 + +	517	[11, 47]

Mixed A323782 and A323784 table for primes < 500

index	prime	BT representation	reversed BT	warped prime	prime factors
1	2	+ -	- +	-2	[2]
2	3	+ 0	0 +	1	[]
3	5	+ - -	- - +	-11	[11]
4	7	+ - +	+ - +	7	[7]
5	11	+ + -	- + +	-5	[5]
6	13	+ + +	+ + +	13	[13]
7	17	+ - 0 -	- 0 - +	-29	[29]
8	19	+ - 0 +	+ 0 - +	25	[5]
9	23	+ 0 - -	- - 0 +	-35	[5, 7]
10	29	+ 0 + -	- + 0 +	-17	[17]
11	31	+ 0 + +	+ + 0 +	37	[37]
12	37	+ + 0 +	+ 0 + +	31	[31]
13	41	+ - - - -	- - - - +	-119	[7, 17]
14	43	+ - - - +	+ - - - +	43	[43]
15	47	+ - - + -	- + - - +	-65	[5, 13]
16	53	+ - 0 0 -	- 0 0 - +	-83	[83]
17	59	+ - + - -	- - + - +	-101	[101]
18	61	+ - + - +	+ - + - +	61	[61]
19	67	+ - + + +	+ + + - +	115	[5, 23]
20	71	+ 0 - 0 -	- 0 - 0 +	-89	[89]
21	73	+ 0 - 0 +	+ 0 - 0 +	73	[73]
22	79	+ 0 0 - +	+ - 0 0 +	55	[5, 11]
23	83	+ 0 0 + -	- + 0 0 +	-53	[53]
24	89	+ 0 + 0 -	- 0 + 0 +	-71	[71]
25	97	+ + - - +	+ - - + +	49	[7]
26	101	+ + - + -	- + - + +	-59	[59]
27	103	+ + - + +	+ + - + +	103	[103]
28	107	+ + 0 0 -	- 0 0 + +	-77	[7, 11]
29	109	+ + 0 0 +	+ 0 0 + +	85	[5, 17]
30	113	+ + + - -	- - + + +	-95	[5, 19]
31	127	+ - - - 0 +	+ 0 - - - +	205	[5, 41]
32	131	+ - - 0 - -	- - 0 - - +	-335	[5, 67]
33	137	+ - - 0 + -	- + 0 - - +	-173	[173]
34	139	+ - - 0 + +	+ + 0 - - +	313	[313]
35	149	+ - 0 - - -	- - - 0 - +	-353	[353]
36	151	+ - 0 - - +	+ - - 0 - +	133	[7, 19]
37	157	+ - 0 - + +	+ + - 0 - +	295	[5, 59]
38	163	+ - 0 0 0 +	+ 0 0 0 - +	241	[241]
39	167	+ - 0 + - -	- - + 0 - +	-299	[13, 23]
40	173	+ - 0 + + -	- + + 0 - +	-137	[137]
41	179	+ - + - 0 -	- 0 - + - +	-263	[263]
42	181	+ - + - 0 +	+ 0 - + - +	223	[223]
43	191	+ - + 0 + -	- + 0 + - +	-155	[5, 31]
44	193	+ - + 0 + +	+ + 0 + - +	331	[331]
45	197	+ - + + 0 -	- 0 + + - +	-209	[11, 19]
46	199	+ - + + 0 +	+ 0 + + - +	277	[277]
47	211	+ 0 - - + +	+ + - - 0 +	289	[17]
48	223	+ 0 - + - +	+ - + - 0 +	181	[181]
49	227	+ 0 - + + -	- + + - 0 +	-143	[11, 13]
50	229	+ 0 - + + +	+ + + - 0 +	343	[7]
51	233	+ 0 0 - 0 -	- 0 - 0 0 +	-269	[269]
52	239	+ 0 0 0 - -	- - 0 0 0 +	-323	[17, 19]
53	241	+ 0 0 0 - +	+ - 0 0 0 +	163	[163]
54	251	+ 0 0 + 0 -	- 0 + 0 0 +	-215	[5, 43]
55	257	+ 0 + - - -	- - - + 0 +	-341	[11, 31]
56	263	+ 0 + - + -	- + - + 0 +	-179	[179]
57	269	+ 0 + 0 0 -	- 0 0 + 0 +	-233	[233]
58	271	+ 0 + 0 0 +	+ 0 0 + 0 +	253	[11, 23]
59	277	+ 0 + + - +	+ - + + 0 +	199	[199]
60	281	+ 0 + + + -	- + + + 0 +	-125	[5]
61	283	+ 0 + + + +	+ + + + 0 +	361	[19]
62	293	+ + - 0 - -	- - 0 - + +	-329	[7, 47]
63	307	+ + - + 0 +	+ 0 + - + +	265	[5, 53]
64	311	+ + 0 - - -	- - - 0 + +	-347	[347]
65	313	+ + 0 - - +	+ - - 0 + +	139	[139]
66	317	+ + 0 - + -	- + - 0 + +	-185	[5, 37]
67	331	+ + 0 + - +	+ - + 0 + +	193	[193]
68	337	+ + 0 + + +	+ + + 0 + +	355	[5, 71]
69	347	+ + + 0 - -	- - 0 + + +	-311	[311]
70	349	+ + + 0 - +	+ - 0 + + +	175	[5, 7]
71	353	+ + + 0 + -	- + 0 + + +	-149	[149]
72	359	+ + + + 0 -	- 0 + + + +	-203	[7, 29]
73	367	+ - - - - - +	+ - - - - - +	367	[367]
74	373	+ - - - - + +	+ + - - - - +	853	[853]
75	379	+ - - - 0 0 +	+ 0 0 - - - +	691	[691]
76	383	+ - - - + - -	- - + - - - +	-929	[929]
77	389	+ - - - + + -	- + + - - - +	-443	[443]
78	397	+ - - 0 - 0 +	+ 0 - 0 - - +	637	[7, 13]
79	401	+ - - 0 0 - -	- - 0 0 - - +	-983	[983]
80	409	+ - - 0 0 + +	+ + 0 0 - - +	961	[31]
81	419	+ - - + - - -	- - - + - - +	-1037	[17, 61]
82	421	+ - - + - - +	+ - - + - - +	421	[421]
83	431	+ - - + 0 0 -	- 0 0 + - - +	-713	[23, 31]
84	433	+ - - + 0 0 +	+ 0 0 + - - +	745	[5, 149]
85	439	+ - - + + - +	+ - + + - - +	583	[11, 53]
86	443	+ - - + + + -	- + + + - - +	-389	[389]
87	449	+ - 0 - - 0 -	- 0 - - 0 - +	-839	[839]
88	457	+ - 0 - 0 - +	+ - 0 - 0 - +	457	[457]
89	461	+ - 0 - 0 + -	- + 0 - 0 - +	-515	[5, 103]
90	463	+ - 0 - 0 + +	+ + 0 - 0 - +	943	[23, 41]
91	467	+ - 0 - + 0 -	- 0 + - 0 - +	-677	[677]
92	479	+ - 0 0 - + -	- + - 0 0 - +	-569	[569]
93	487	+ - 0 0 0 0 +	+ 0 0 0 0 - +	727	[727]
94	491	+ - 0 0 + - -	- - + 0 0 - +	-893	[19, 47]
95	499	+ - 0 0 + + +	+ + + 0 0 - +	1051	[1051]

Relation between balanced ternary operation of A117966 and warp operator A323782 on integers

Python Code from A117966: Balanced ternary enumeration (or balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s

def a(n):
    if n==0: return 0
    if n%3==0: return 3*a(n/3)
    elif n%3==1: return 3*a((n - 1)/3) + 1
    else: return 3*a((n - 2)/3) - 1`

Python Code from A134028: Reversal of n in balanced ternary representation (broken)

def a(n):
    if n==0: return 0
    s=[]
    x=0
    while n>0:
        x=n%3
        n=n/3
        if x==2:
            x=-1
            n+=1
        s+=[x, ]
    l=s[::-1]
    t=0
    for i in xrange(len(l)): t+=l[i]*3**i
    return t

Comparative table between A117966 & A134028

index	A117966 operator	A323782 & A134028 warp operator
1	1	1
2	-1	-2
3	3	1
4	4	4
5	2	-11
6	-3	-2
7	-2	7
8	-4	-8
9	9	1
10	10	10
11	8	-5
12	12	4
13	13	13
14	11	-38
15	6	-11
16	7	16
17	5	-29
18	-9	-2
19	-8	25
20	-10	-20
21	-6	7
22	-5	34
23	-7	-35
24	-12	-8
25	-11	19
26	-13	-26
27	27	1
28	28	28
29	26	-17
30	30	10
31	31	37
32	29	-32
33	24	-5
34	25	22
35	23	-23
36	36	4
37	37	31
38	35	-14
39	39	13
40	40	40
41	38	-119
42	33	-38
43	34	43
44	32	-92
45	18	-11
46	19	70
47	17	-65
48	21	16
49	22	97
50	20	-110
51	15	-29
52	16	52
53	14	-83
54	-27	-2
55	-26	79
56	-28	-56
57	-24	25
58	-23	106
59	-25	-101
60	-30	-20
61	-29	61
62	-31	-74
63	-18	7
64	-17	88
65	-19	-47
66	-15	34
67	-14	115
68	-16	-116
69	-21	-35
70	-20	46
71	-22	-89
72	-36	-8
73	-35	73
74	-37	-62
75	-33	19
76	-32	100
77	-34	-107
78	-39	-26
79	-38	55
80	-40	-80
81	81	1
82	82	82
83	80	-53
84	84	28
85	85	109
86	83	-98
87	78	-17
88	79	64
89	77	-71
90	90	10
91	91	91
92	89	-44
93	93	37
94	94	118
95	92	-113
96	87	-32
97	88	49
98	86	-86
99	72	-5
100	73	76