Other Special Classes of Interprimes

On the 1st November 2023, I posted on A Special Class of Interprime and these were non-palindromic composite numbers located between twin primes which, when reversed, are also located between twin primes. Some work both ways while some are only one way because they end in a zero. Figure 1 shows an example of the former while Figure 2 shows an example of the latter.

Figure 1

Figure 2

Today I turned 27738 days old and this number is an interprime number between twin primes which when concatenated with itself forms a number which is also an interprime between twin primes. The result for 27738 is shown in Figure 3.

Figure 3

Numbers of this sort belong to OEIS A235109 :

A235109     Averages q of twin prime pairs, such that q concatenated to q is also the average of a twin prime pair.

The initial members are (permalink):

42, 102, 108, 180, 192, 270, 312, 420, 522, 660, 822, 882, 1230, 1482, 4242, 4788, 8820, 10332, 11550, 13692, 14550, 14562, 14868, 15732, 17910, 18522, 20550, 21648, 22620, 23670, 23832, 26262, 27738, 35838, 38922, 39042, 40128, 42018, 43962, 44532, 46440

As a variation on this, we could concatenate an interprime with its reversal, thus forming a palindrome. This is shown in Figure 4.

Figure 4

Up to 40000, the initial interprimes with this property are (permalink) 240, 270, 2142, 8388, 22092, 22962, 23832, 24420, 24918, 26262, 27690 and 28110. The sequence does not appear in the OEIS. The members of this sequence are, to be fair, rather sparse and could be made more numerous if the condition that the interprime lay between twin primes was relaxed. If we simply require that the interprime, when concatenated with its reverse, is also an interprime then in the range up to 40000, 263 numbers satisfy. The numbers are (permalink):

9, 15, 21, 42, 93, 102, 105, 108, 160, 240, 246, 270, 279, 324, 386, 432, 754, 810, 909, 933, 1092, 1302, 1452, 1611, 1998, 2142, 2205, 2295, 2322, 2336, 2470, 2568, 2667, 2892, 2900, 2946, 3021, 3326, 3423, 3453, 3465, 3558, 3588, 3627, 3672, 3736, 3885, 3921, 4002, 4065, 4076, 4131, 4353, 4422, 4646, 4742, 4785, 5193, 5439, 5481, 5502, 5529, 5607, 5804, 6107, 6340, 6376, 6798, 6969, 7182, 7212, 7494, 8097, 8169, 8388, 8437, 8844, 8908, 8985, 9394, 9678, 9865, 10008, 10101, 10794, 10815, 10875, 10944, 10998, 11226, 11445, 11523, 11817, 12024, 12111, 12252, 12489, 12500, 12514, 12826, 12947, 13056, 13101, 13320, 13374, 13482, 13560, 13674, 13740, 13881, 13965, 14064, 14415, 14592, 14715, 15015, 15087, 15534, 15664, 16230, 16396, 16799, 17388, 17529, 17958, 18042, 18288, 18360, 18447, 18531, 18737, 19149, 19314, 19548, 19704, 19857, 20022, 20049, 20057, 20225, 20358, 20403, 20687, 20745, 20751, 20808, 21015, 21104, 21189, 21202, 21381, 21404, 21558, 21969, 22092, 22272, 22719, 22866, 22904, 22962, 23124, 23631, 23832, 24036, 24144, 24333, 24420, 24522, 24804, 24855, 24918, 25001, 25080, 25305, 25417, 25455, 25470, 25578, 25595, 25761, 25932, 25960, 26180, 26262, 26412, 26582, 26637, 26675, 26748, 27075, 27429, 27546, 27597, 27690, 27999, 28110, 28117, 28253, 28314, 28410, 28629, 28692, 28869, 29247, 29577, 29720, 29826, 29865, 29937, 30106, 30165, 30217, 30270, 30693, 31149, 31152, 31182, 31269, 31536, 31617, 31653, 31977, 32244, 32325, 32700, 32914, 33186, 33288, 33573, 33588, 33621, 33639, 33854, 33939, 34125, 34290, 34412, 34590, 34683, 34743, 34874, 34962, 35094, 35421, 35674, 35802, 36003, 36442, 36486, 36648, 36694, 36764, 37220, 37514, 37548, 38385, 38856, 39093, 39159, 39447, 39627, 39852, 39999

Let's take 93 from the previous list as an example. It is an interprime that lies midway between 89 and 97. Concatenated with its reverse (39), we get 9339 and this number is midway between 9337 and 9341. Figure 5 illustrates this.

Figure 5

Similarly we could relax the interprime condition for interprimes that are concatenated with themselves (but not reversed). There are 345 interprimes in the range up to 40000 that qualify. An example is 21, an interprime between 19 and 23, that forms 2121, an interprime between 2113 and 2129.

Figure 6

The interprimes between 27700 and 40000 with this property are (permalink):

..., 27738, 27888, 27945, 27990, 28281, 28515, 28613, 28740, 28815, 28851, 28994, 29013, 29170, 29237, 29307, 29448, 29835, 29953, 30000, 30038, 30264, 30300, 30310, 30378, 30468, 30555, 30846, 30856, 30902, 31080, 31122, 31269, 31335, 31347, 31660, 31854, 31960, 32298, 32361, 32715, 32925, 32990, 33018, 33235, 33351, 33465, 33594, 33717, 33840, 33860, 34224, 34734, 34743, 34848, 35325, 35556, 35571, 35838, 35980, 36189, 36462, 36680, 37008, 37053, 37176, 37576, 37850, 38076, 38238, 38310, 38331, 38685, 38922, 39042, 39084, 39093, 39105, 39363, 39378, 39447, 39546, 39691, 39765, 39774, 39894, ...

Lastly, if we relax the interprime condition that the interprime and its reverse must lie between twin primes, then there are 629 numbers that satisfy in the range up to 40000 (permalink) but I won't list those here.