|
%I #25 Aug 03 2016 09:01:49
%S 1,3,5,15,1,3,9,27,1,5,7,35,1,3,13,39,1,3,17,51,1,5,11,55,1,3,7,9,21,
%T 63,1,3,5,15,25,75,1,3,29,87,1,7,13,91,1,5,19,95,1,3,9,11,33,99,1,3,
%U 37,111,1,5,23,115,1,7,17,119,1,3,41,123,1,3,5,9,15,27,45,135,1,11,13,143,1,3,7,21,49,147,1,5,31,155
%N Irregular array read by rows: T(n, k) giving in row n the divisors of nonprime numbers that are 3 (mod 4).
%C The length of row n is 2*A273165(n).
%C The number of divisors 1 and -1 (mod 4) in each row are identical, namely A273165(n). See the Jan 05 2004 Jovovic comment on A078703. For prime numbers 3 (mod 4) this is obvious. For the proof see a comment on A091236 with the Grosswald reference.
%C From _Paul Curtz_, Jul 31 2016: (Start)
%C For each row n of length 2*r(n) one has:
%C T(n, m)*T(n, 2*r(n)-m+1) = T(n, 2*r(n)),for m=1, 2, ... , r(n).
%C From the second comment it follows that the row sums are congruent to 0 modulo 4. (End)
%F T(n, k) gives the k-th divisor of A091236(n) in increasing order.
%e The irregular array T(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 ...
%e 1: 1 3 5 15
%e 2: 1 3 9 27
%e 3: 1 5 7 35
%e 4: 1 3 13 39
%e 5: 1 3 17 51
%e 6: 1 5 11 55
%e 7: 1 3 7 9 21 63
%e 8: 1 3 5 15 25 75
%e 9: 1 3 29 87
%e 10: 1 7 13 91
%e 11: 1 5 19 95
%e 12: 1 3 9 11 33 99
%e 13: 1 3 37 111
%e 14: 1 5 23 115
%e 15: 1 7 17 119
%e 16: 1 3 41 123
%e 17: 1 3 5 9 15 27 45 135
%e 18: 1 11 13 143
%e 19: 1 3 7 21 49 147
%e 20: 1 5 31 155
%e ...
%e The irregular array modulo 4 gives (-1 for 3 (mod 4)):
%e n\k 1 2 3 4 5 6 7 8 ...
%e 1: 1 -1 1 -1
%e 2: 1 -1 1 -1
%e 3: 1 1 -1 -1
%e 4: 1 -1 1 -1
%e 5: 1 -1 1 -1
%e 6: 1 1 -1 -1
%e 7: 1 -1 1 -1 1 -1
%e 8: 1 -1 1 -1 1 -1
%e 9: 1 -1 1 -1
%e 10: 1 -1 1 -1
%e 11: 1 1 -1 -1
%e 12: 1 -1 1 -1 1 -1
%e 13: 1 -1 1 -1
%e 14: 1 1 -1 -1
%e 15: 1 -1 1 -1
%e 16: 1 -1 1 -1
%e 17: 1 -1 1 1 -1 -1 1 -1
%e 18: 1 -1 1 -1
%e 19: 1 -1 -1 1 1 -1
%e 20: 1 1 -1 -1
%e ...
%t Divisors@ Select[Range@ 155, CompositeQ@ # && Mod[#, 4] == 3 &] // Flatten (* _Michael De Vlieger_, Aug 01 2016 *)
%Y Cf. A004767, A078703, A091236, A273165.
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Jul 29 2016
|
