A273165 -id:A273165 - cp's OEIS frontend

A273164 Irregular array read by rows: T(n, k) giving in row n the divisors of nonprime numbers that are 3 (mod 4).

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, 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, 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

Offset: 1

Wolfdieter Lang, Jul 29 2016

The length of row n is 2*A273165(n).
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.
From Paul Curtz, Jul 31 2016: (Start)
For each row n of length 2*r(n) one has:
T(n, m)*T(n, 2*r(n)-m+1) = T(n, 2*r(n)),for m=1, 2, ... , r(n).
From the second comment it follows that the row sums are congruent to 0 modulo 4. (End)

			The irregular array T(n, k) begins:
n\k 1  2  3   4  5   6  7   8 ...
1:  1  3  5  15
2:  1  3  9  27
3:  1  5  7  35
4:  1  3 13  39
5:  1  3 17  51
6:  1  5 11  55
7:  1  3  7   9 21  63
8:  1  3  5  15 25  75
9:  1  3 29  87
10: 1  7 13  91
11: 1  5 19  95
12: 1  3  9  11 33  99
13: 1  3 37 111
14: 1  5 23 115
15: 1  7 17 119
16: 1  3 41 123
17: 1  3  5   9 15  27 45 135
18: 1 11 13 143
19: 1  3  7  21 49 147
20: 1 5 31 155
...
The irregular array modulo 4 gives (-1 for 3 (mod 4)):
n\k 1  2  3   4  5   6  7   8 ...
1:  1  -1  1  -1
2:  1  -1  1  -1
3:  1   1 -1  -1
4:  1  -1  1  -1
5:  1  -1  1  -1
6:  1   1 -1  -1
7:  1  -1  1  -1  1  -1
8:  1  -1  1  -1  1  -1
9:  1  -1  1  -1
10: 1  -1  1  -1
11: 1   1 -1  -1
12: 1  -1  1  -1  1  -1
13: 1  -1  1  -1
14: 1   1 -1  -1
15: 1  -1  1  -1
16: 1  -1  1  -1
17: 1  -1  1   1  -1  -1  1 -1
18: 1  -1  1  -1
19: 1  -1 -1   1   1  -1
20: 1   1 -1  -1
...

Cf. A004767, A078703, A091236, A273165.

Divisors@ Select[Range@ 155, CompositeQ@ # && Mod[#, 4] == 3 &] // Flatten (* Michael De Vlieger, Aug 01 2016 *)

T(n, k) gives the k-th divisor of A091236(n) in increasing order.

cp's OEIS Frontend

A273164 Irregular array read by rows: T(n, k) giving in row n the divisors of nonprime numbers that are 3 (mod 4).

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