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Warp Primes generator for OEIS
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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.

Ulam Spiral of A323782 and A323784

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.

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
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