Biprime Prime Time

For some reason, it only just occurred to me that I can search this blog for the occurrence of particular numbers. Usually I search the OEIS first and if nothing of interest comes up, I search my Bespoken for Sequences database. If there's nothing there, I'll search the airtable.com database and if nothing turns up, I'll search for the number in the OEIS b-files. However, searching this blog should probably be my second priority if nothing comes up in the OEIS. After all I have hundreds of posts and thousands of numbers.

The theme of this post concerns biprimes or semiprimes where there is a connection between one prime and the other. The idea derived from the number associated with my diurnal age today: 27149. This number is a member of OEIS A045925:

A045925

a(\(n\)) = \(n\) * Fibonacci(\(n\)).

When \(n=17\), we have Fibonacci(17) = 1597 which is prime and 27149 = 17 x 1597. Very few of the members of this sequence are biprimes because \(n\) can be composite and\or Fibonacci(\(n\)) can be composite. The biprimes are listed below (in bold):

3 --> 6 = 2 * 3
5 --> 25 = 5^2
7 --> 91 = 7 * 13
11 --> 979 = 11 * 89
13 --> 3029 = 13 * 233
17 --> 27149 = 17 * 1597
23 --> 659111 = 23 * 28657
29 --> 14912641 = 29 * 514229
43 --> 18640260791 = 43 * 433494437
47 --> 139647108431 = 47 * 2971215073

We are looking for biprimes of the form prime multiplied by some function of that prime where the function generates a new prime. An easy source of such numbers arises from OEIS A073065: 

A073065

a(\(n\)) = prime(\(n\)) * prime(prime(\(n\))).

The sequence begins: 6, 15, 55, 119, 341, 533, 1003, 1273, 1909, 3161, 3937, 5809, 7339, 8213, 9917, 12773, 16343, 17263, 22177, 25063, 26791, 31679, 35773, 41029

The breakdown is as follows:

1 --> 2 x 3 = 6

2 --> 3 x 5 = 15

3 --> 5 x 11 = 55

4 --> 7 x 17 = 119

5 --> 11 x 31 = 341

6 --> 13 x 41 = 533

7 --> 17 x 59 = 1003

8 --> 19 x 67 = 1273

9 --> 23 x 83 = 1909

10 --> 29 x 109 = 3161

11 --> 31 x 127 = 3937

12 --> 37 x 157 = 5809

13 --> 41 x 179 = 7339

14 --> 43 x 191 = 8213

15 --> 47 x 211 = 9917

16 --> 53 x 241 = 12773

17 --> 59 x 277 = 16343

18 --> 61 x 283 = 17263

19 --> 67 x 331 = 22177

20--> 71 x 353 = 25063

21 --> 73 x 367 = 26791

22 --> 79 x 401 = 31679

23 --> 83 x 431 = 35773

24 --> 89 x 461 = 41029

25 --> 97 x 509 = 49373

Another way to find "special" biprimes is to determine which of them concatenate to form new primes, either by smaller with larger or larger with smaller. Such biprimes form OEIS A330441:

A330441

Semiprimes \(p \times q\) such that the concatenations of \(p\) and \(q\) in both orders are prime.

Here is a list of the initial members (508) of this sequence up to 40000:

21, 33, 51, 93, 111, 133, 177, 201, 219, 247, 253, 327, 411, 427, 573, 589, 679, 687, 763, 793, 813, 889, 993, 1077, 1081, 1119, 1243, 1339, 1347, 1401, 1411, 1497, 1501, 1603, 1623, 1651, 1671, 1821, 1839, 1843, 1851, 1981, 2019, 2047, 2059, 2103, 2157, 2199, 2217, 2469, 2479, 2629, 2761, 2787, 2841, 2923, 3031, 3039, 3057, 3097, 3099, 3133, 3153, 3409, 3439, 3543, 3579, 3661, 3711, 3787, 3829, 3883, 3973, 4063, 4171, 4303, 4309, 4369, 4381, 4429, 4443, 4593, 4681, 4711, 4771, 4821, 4837, 4843, 4881, 4971, 4989, 5001, 5097, 5191, 5299, 5533, 5611, 5721, 5761, 5803, 5971, 5989, 6181, 6207, 6429, 6457, 6511, 6559, 6613, 6639, 6799, 6891, 7003, 7131, 7143, 7153, 7233, 7323, 7327, 7363, 7501, 7509, 7633, 7711, 7737, 8121, 8173, 8197, 8227, 8347, 8367, 8529, 8659, 8743, 8751, 8871, 9147, 9211, 9217, 9229, 9307, 9313, 9357, 9493, 9523, 9589, 9793, 9853, 9903, 10063, 10171, 10249, 10297, 10351, 10383, 10407, 10441, 10483, 10519, 10537, 10777, 10807, 10843, 10849, 10911, 10963, 11107, 11307, 11539, 11581, 11623, 11629, 11653, 11707, 11733, 11769, 11793, 11797, 11851, 12001, 12057, 12133, 12187, 12193, 12477, 12643, 12759, 12961, 13027, 13117, 13153, 13387, 13471, 13749, 13771, 13813, 13951, 13993, 14187, 14257, 14623, 14677, 14757, 14803, 14977, 15049, 15127, 15151, 15153, 15177, 15229, 15247, 15297, 15513, 15571, 15697, 15769, 15843, 15883, 15969, 16003, 16009, 16143, 16257, 16347, 16357, 16387, 16501, 16507, 16521, 16543, 16563, 16593, 16719, 16837, 17089, 17131, 17173, 17269, 17371, 17403, 17503, 17517, 17521, 17607, 17769, 17779, 17803, 17833, 17953, 18111, 18219, 18247, 18319, 18673, 18697, 18709, 18789, 18829, 18831, 18841, 18871, 18943, 18949, 19021, 19033, 19039, 19059, 19353, 19357, 19407, 19587, 19627, 19689, 19693, 19713, 19741, 19797, 19807, 19897, 19911, 19939, 19959, 20059, 20191, 20221, 20317, 20379, 20581, 20613, 20671, 20697, 20701, 20841, 21117, 21133, 21151, 21171, 21253, 21439, 21457, 21477, 21691, 21703, 21759, 22069, 22107, 22207, 22267, 22459, 22507, 22759, 22773, 22947, 23073, 23317, 23377, 23503, 23527, 23713, 23731, 23889, 24031, 24193, 24313, 24343, 24487, 24607, 24613, 24643, 24657, 24711, 24721, 24949, 25081, 25131, 25267, 25291, 25293, 25507, 25549, 25651, 25729, 25777, 25813, 25887, 25963, 26007, 26077, 26089, 26097, 26139, 26167, 26241, 26329, 26349, 26359, 26401, 26629, 26761, 26989, 27049, 27309, 27469, 27471, 27493, 27543, 27619, 27723, 27781, 28117, 28129, 28189, 28383, 28417, 28563, 28831, 28869, 28993, 29113, 29227, 29239, 29407, 29479, 29533, 29571, 29647, 29661, 29703, 29707, 29713, 29731, 30273, 30333, 30499, 30507, 30511, 30607, 30669, 30721, 30729, 30799, 30979, 30999, 31071, 31087, 31273, 31279, 31363, 31377, 31453, 31459, 31483, 31549, 31621, 31701, 31711, 31881, 31969, 32167, 32197, 32281, 32313, 32667, 32743, 32961, 33043, 33081, 33103, 33519, 33531, 33591, 33913, 34063, 34117, 34179, 34321, 34339, 34341, 34417, 34459, 34609, 34653, 34789, 34813, 34933, 34957, 35173, 35299, 35359, 35383, 35463, 35481, 35601, 35691, 35743, 35779, 35943, 35953, 36019, 36181, 36213, 36259, 36303, 36367, 36601, 36609, 36649, 36667, 36759, 36763, 36769, 36807, 36961, 37029, 37063, 37099, 37239, 37267, 37351, 37353, 37399, 37621, 37803, 38059, 38109, 38131, 38137, 38347, 38359, 38389, 38523, 38623, 38647, 38683, 38697, 38929, 38937, 39007, 39049, 39427, 39487, 39553, 39613, 39649, 39823, 39931, 39967

 For example, 27309= 3 * 9103 and the concatenations 39103 and 91033 are both prime.