cp's OEIS Frontend

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.

%I A306388 #76 Nov 12 2019 09:01:19
%S A306388 1,3,5,13,17,61,65,213,329,885,1025,3933,4097,13781,22121,54613,65537,
%T A306388 251741,262145,906613,1364681,3497301,4194305,16111453,17859617,
%U A306388 55932245,86282825,225793493,268435457,1064687485,1073741825,3579139413,5526297161,14316688725
%N 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.
%C A306388 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).
%C A306388 a(n) -in binary, and recorded as a triangle- gives a 'Totient map' for the naturals.
%C A306388    1                                    1
%C A306388    2                                   11
%C A306388    3                                  101
%C A306388    4                                 1101
%C A306388    5                                10001
%C A306388    6                               111101
%C A306388    7                              1000001
%C A306388    8                             11010101
%C A306388    9                            101001001
%C A306388   10                           1101110101
%C A306388   11                          10000000001
%C A306388   12                         111101011101
%C A306388   13                        1000000000001
%C A306388   14                       11010111010101
%C A306388   15                      101011001101001
%C A306388   16                     1101010101010101
%C A306388   ...
%H A306388 Amiram Eldar, <a href="/A306388/b306388.txt">Table of n, a(n) for n = 1..1000</a>
%e A306388 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.
%e A306388 a(c), c composite, is computable as follows:
%e A306388 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* =
%e A306388         110101
%e A306388       + 101001
%e A306388        _______
%e A306388         111101, which, when reverted to decimal, gives a(6).
%e A306388 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* =
%e A306388         110101010101
%e A306388       + 101001001001
%e A306388        ______________
%e A306388         111101011101, which, when reverted to decimal, gives a(12).
%e A306388 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* =
%e A306388     110101010101010101010101010101
%e A306388     101001001001001001001001001001
%e A306388   + 100010000100001000010000100001
%e A306388     ______________________________
%e A306388     111111011101011101011101111101, which, when reverted to decimal, gives a(30).
%t A306388 a[n_] := FromDigits[Boole@(#==1 || GCD[#,n] > 1) &/@ Range[n], 2]; Array[a, 30] (* _Amiram Eldar_, Mar 26 2019 *)
%o A306388 (PARI) a(n) = my(v=vector(n, k, if (k==1, 1, gcd(k, n) != 1))); fromdigits(v, 2); \\ _Michel Marcus_, Mar 28 2019
%Y A306388 Cf. A054432.
%K A306388 nonn,base
%O A306388 1,2
%A A306388 _Christopher Hohl_, Mar 01 2019
%E A306388 More terms from _Amiram Eldar_, Mar 26 2019
%E A306388 Name clarified by _Michel Marcus_, Mar 28 2019