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 A185156 Primes with the property that complementing any two different bits in the binary representation of these primes never produces a prime number. 0
 2, 3, 2731, 174763, 715827883, 1464948053 (list; graph; refs; listen; history; text; internal format)
 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       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

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Last modified April 20 01:55 EDT 2021. Contains 343118 sequences. (Running on oeis4.)