A185156 Primes with the property that complementing any two different bits in the binary representation of these primes never produces a prime number.

2, 3, 2731, 174763, 715827883, 1464948053

Offset

1,1

Comments

Also called weakly primes of 2nd order in base 2.

Formal definition: let P = set of prime numbers, XOR(x,y) = bitwise x xor y, set of witnesses for an integer x>1 w(x) := Union_{1<=k<=floor(log_2(x)), 0<=j<k}{XOR(x, 2^k+2^j)}; then a is in the sequence iff (a in P)&( Intersection(w(a), P) = {}).

There are only 6 terms < 10^11 (exhaustive search). But several larger terms of a special form are known (Wagstaff primes, A000979). The smallest of them are:

a(6+)=2932031007403,

a(7+)=768614336404564651,

a(8+)=201487636602438195784363. - Terentyev Oleg

Links

Table of n, a(n) for n=1..6.

Example

a(3)=2731 is in the sequence because it is prime and all its witnesses are composite numbers :
2731  =  101010101011 ->       10101011  =     171  =  3^2 * 19
                             1000101011  =     555  =  3 * 5 * 37
                             1010001011  =     651  =  3 * 7 * 31
                             1010100011  =     675  =  3^3 * 5^2
                             1010101001  =     681  =  3 * 227
                             1010101010  =     682  =  2 * 11 * 31
                             1010101111  =     687  =  3 * 229
                             1010111011  =     699  =  3 * 233
                             1011101011  =     747  =  3^2 * 83
                             1110101011  =     939  =  3 * 313
                            11010101011  =    1707  =  3 * 569
                           100000101011  =    2091  =  3 * 17 * 41
                           100010001011  =    2187  =  3^7
                           100010100011  =    2211  =  3 * 11 * 67
                           100010101001  =    2217  =  3 * 739
                           100010101010  =    2218  =  2 * 1109
                           100010101111  =    2223  =  3^2 * 13 * 19
                           100010111011  =    2235  =  3 * 5 * 149
                           100011101011  =    2283  =  3 * 761
                           100110101011  =    2475  =  3^2 * 5^2 * 11
                           101000001011  =    2571  =  3 * 857
                           101000100011  =    2595  =  3 * 5 * 173
                           101000101001  =    2601  =  3^2 * 17^2
                           101000101010  =    2602  =  2 * 1301
                           101000101111  =    2607  =  3 * 11 * 79
                           101000111011  =    2619  =  3^3 * 97
                           101001101011  =    2667  =  3 * 7 * 127
                           101010000011  =    2691  =  3^2 * 13 * 23
                           101010001001  =    2697  =  3 * 29 * 31
                           101010001010  =    2698  =  2 * 19 * 71
                           101010001111  =    2703  =  3 * 17 * 53
                           101010011011  =    2715  =  3 * 5 * 181
                           101010100001  =    2721  =  3 * 907
                           101010100010  =    2722  =  2 * 1361
                           101010100111  =    2727  =  3^3 * 101
                           101010101000  =    2728  =  2^3 * 11 * 31
                           101010101101  =    2733  =  3 * 911
                           101010101110  =    2734  =  2 * 1367
                           101010110011  =    2739  =  3 * 11 * 83
                           101010111001  =    2745  =  3^2 * 5 * 61
                           101010111010  =    2746  =  2 * 1373
                           101010111111  =    2751  =  3 * 7 * 131
                           101011001011  =    2763  =  3^2 * 307
                           101011100011  =    2787  =  3 * 929
                           101011101001  =    2793  =  3 * 7^2 * 19
                           101011101010  =    2794  =  2 * 11 * 127
                           101011101111  =    2799  =  3^2 * 311
                           101011111011  =    2811  =  3 * 937
                           101100101011  =    2859  =  3 * 953
                           101110001011  =    2955  =  3 * 5 * 197
                           101110100011  =    2979  =  3^2 * 331
                           101110101001  =    2985  =  3 * 5 * 199
                           101110101010  =    2986  =  2 * 1493
                           101110101111  =    2991  =  3 * 997
                           101110111011  =    3003  =  3 * 7 * 11 * 13
                           101111101011  =    3051  =  3^3 * 113
                           110010101011  =    3243  =  3 * 23 * 47
                           111000101011  =    3627  =  3^2 * 13 * 31
                           111010001011  =    3723  =  3 * 17 * 73
                           111010100011  =    3747  =  3 * 1249
                           111010101001  =    3753  =  3^3 * 139
                           111010101010  =    3754  =  2 * 1877
                           111010101111  =    3759  =  3 * 7 * 179
                           111010111011  =    3771  =  3^2 * 419
                           111011101011  =    3819  =  3 * 19 * 67
                           111110101011  =    4011  =  3 * 7 * 191

Mathematica

isWPof2ndOrderBase2[x_] := Module[{j = 1, k = 2, flag = x <= 3 || ! BitAnd[x - 3, x - 4] == 0, bitlen = BitLength@x}, While[flag && k < bitlen, While[flag && j < k, flag = !PrimeQ@BitXor[x, BitShiftLeft[1, j] + BitShiftLeft[1, k]]; j++]; j = 1; k++]; flag]; Select[Prime[Range[20000]], isWPof2ndOrderBase2]

Crossrefs

Sequence in context: A066848 A324310 A125612 * A235935 A182383 A257552

Adjacent sequences: A185153 A185154 A185155 * A185157 A185158 A185159

Keyword

nonn,base

Author

Terentyev Oleg, Dec 22 2011

Status

approved