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 A306388 a(n) is a decimal number k having a length n binary expansion which encodes, from left to right at digit j, the coprimality (0) or non-coprimality (1) of j to n, for 1 < j <= n, except for the first digit, which is always 1. 1
 1, 3, 5, 13, 17, 61, 65, 213, 329, 885, 1025, 3933, 4097, 13781, 22121, 54613, 65537, 251741, 262145, 906613, 1364681, 3497301, 4194305, 16111453, 17859617, 55932245, 86282825, 225793493, 268435457, 1064687485, 1073741825, 3579139413, 5526297161, 14316688725 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let Sum* be a special summation procedure carried out on the binary expansions of each of the decimal values produced by the following expression for all distinct prime factors of n. That is, when 'adding' the various binary expansions of said decimal results for each p dividing n, p prime, allow that 1 + q + r + ... + s = 1, and 0 + 0 + ... + 0 = 0. Then, Sum*_{p|n} 2^(p-1) * ((2^p+1) * 2^(n-p) - 2)/(2^p - 1) + 1, when reverted to decimal, gives a(n). a(n) -in binary, and recorded as a triangle- gives a 'Totient map' for the naturals. 1 1 2 11 3 101 4 1101 5 10001 6 111101 7 1000001 8 11010101 9 101001001 10 1101110101 11 10000000001 12 111101011101 13 1000000000001 14 11010111010101 15 101011001101001 16 1101010101010101 ... LINKS Amiram Eldar, Table of n, a(n) for n = 1..1000 EXAMPLE a(p), p prime, are always 2^(p-1)+1, a result of ((2^p+1)*2^(n-p)-2)/(2^p-1)- the main parenthetical term in Sum*- being equal to 1. a(c), c composite, is computable as follows: a(6) = 61 because 6 has the distinct prime factors 2 and 3. So, the special summation of 2^(2-1) * ((2^2 + 1) * 2^(6-2) - 2)/(2^2 - 1) + 1 = 53, a decimal number which has a length 6 binary expansion (110101), and 2^(3-1) * ((2^3 + 1) * 2^(6-3) - 2)/(2^3 - 1) + 1 = 41, another decimal number which has a length 6 binary expansion (101001), gives Sum* = 110101 + 101001 _______ 111101, which, when reverted to decimal, gives a(6). a(12) = 3933 because 12 has the distinct prime factors 2 and 3. So, the special summation of 2^(2-1) * ((2^2 + 1) * 2^(12-2) - 2)/(2^2 - 1) + 1 = 3413, a decimal number which has a length 12 binary expansion (110101010101), and 2^(3-1) * ((2^3 + 1) * 2^(12-3) - 2)/(2^3 - 1) + 1 = 2633, another decimal number which has a length 12 binary expansion (101001001001), gives Sum* = 110101010101 + 101001001001 ______________ 111101011101, which, when reverted to decimal, gives a(12). Likewise, a(30) = 1064687485 because 30 has the distinct prime factors 2, 3, and 5. So, the special summation of 2^(2-1) * ((2^2 + 1) * 2^(30-2) - 2)/(2^2 - 1) + 1 = 894784853 = 110101010101010101010101010101 (length 30), and 2^(3-1) *((2^3 + 1) * 2^(30-3) - 2)/(2^3 - 1) + 1 = 690262601 = 101001001001001001001001001001, and 2^(5-1) * ((2^5 + 1) * 2^(30-5) - 2)/(2^5 - 1) + 1 = 571507745 = 100010000100001000010000100001, gives Sum* = 110101010101010101010101010101 101001001001001001001001001001 + 100010000100001000010000100001 ______________________________ 111111011101011101011101111101, which, when reverted to decimal, gives a(30). MATHEMATICA a[n_] := FromDigits[Boole@(#==1 || GCD[#, n] > 1) &/@ Range[n], 2]; Array[a, 30] (* Amiram Eldar, Mar 26 2019 *) PROG (PARI) a(n) = my(v=vector(n, k, if (k==1, 1, gcd(k, n) != 1))); fromdigits(v, 2); \\ Michel Marcus, Mar 28 2019 CROSSREFS Cf. A054432. Sequence in context: A038185 A284241 A284305 * A131020 A283398 A084706 Adjacent sequences: A306385 A306386 A306387 * A306389 A306390 A306391 KEYWORD nonn,base AUTHOR Christopher Hohl, Mar 01 2019 EXTENSIONS More terms from Amiram Eldar, Mar 26 2019 Name clarified by Michel Marcus, Mar 28 2019 STATUS approved

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Last modified June 5 17:07 EDT 2023. Contains 363138 sequences. (Running on oeis4.)