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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 February 20 13:20 EST 2020. Contains 332077 sequences. (Running on oeis4.)