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
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Comments
Examples
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 * 191Programs
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]