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

1,2

Comments

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)

Links

Table of n, a(n) for n=1..92.

Formula

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

Example

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
...

Mathematica

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

Crossrefs

Cf. A004767, A078703, A091236, A273165.

Sequence in context: A167553 A370612 A118562 * A317661 A339972 A115043

Adjacent sequences: A273161 A273162 A273163 * A273165 A273166 A273167

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Jul 29 2016

Status

approved