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A273164 Irregular array read by rows: T(n, k) giving in row n the divisors of nonprime numbers that are 3 (mod 4). 1
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 (list; graph; refs; listen; history; text; internal format)
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: A129326 A167553 A118562 * A317661 A115043 A272255

Adjacent sequences:  A273161 A273162 A273163 * A273165 A273166 A273167

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Jul 29 2016

STATUS

approved

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Last modified March 19 00:03 EDT 2019. Contains 321305 sequences. (Running on oeis4.)