PRIME + and NON-PRIME - Improved Format

As with the EVEN + and ODD - and EVEN - and ODD + recursive algorithms, I've taken the original 393 page data document and converted it to a 83 page PDF that is more reader friendly. Here some excerpts.

Figure 1

Figure 2

Figure 3

Table 1

Table 2

Table 3

Table 4

Some Categories of Primes

There is a category of prime numbers with the property that when both the sum of their digits and the product of their digits is added to the number then the new, resultant numbers are also prime. An example would be 28181 with a sum of digits of 20 and a product of digits of 128 where:$$ \begin{align} 28181 + 20 &= 28201 \text{ prime} \\ 28181 + 128 &= 28309 \text{ prime} \end{align}$$In the range up to 40000, these primes have a density of 7.376% compared to all primes. Here is a list of such primes between 28000 and 40000 (permalink):

28097, 28181, 28703, 28901, 29153, 29179, 29209, 30089, 30119, 30203, 30313, 30449, 30469, 30539, 30557, 30649, 30661, 30713, 30803, 30809, 30829, 31019, 31307, 32063, 32069, 32083, 32173, 32203, 32401, 32687, 32957, 32971, 33013, 33037, 33091, 33301, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35053, 35059, 35251, 35257, 35323, 35507, 35509, 35521, 35569, 35831, 36209, 36229, 36469, 36559, 36607, 36919, 37019, 37039, 37097, 37321, 37369, 37501, 37507, 37547, 37871, 38047, 38351, 38959, 39019, 39079, 39103, 39161, 39301, 39521

These primes constitute OEIS A128717:

A128717: primes that yield another prime if one adds either the sum of its digits or the product of its digits.

Another category of prime involves its cube being pandigital, meaning that each digit from 0 to 9 occurs at least once with duplicates being permitted. Again 28181 satisfies this condition:$$28181^3 = 20753798525641$$Primes of this sort constitute:

A124629: primes \(p\) such that their cubes are pandigital.

The members of this sequence up to 40000 have a density is 1.523 % compared to all primes and these are (permalink):

5437, 6221, 7219, 8443, 10903, 11353, 15937, 17123, 18229, 19429, 20353, 20903, 20929, 21803, 21841, 21961, 22123, 22283, 22993, 23053, 23369, 23663, 24733, 25183, 25219, 25463, 26317, 26387, 26449, 27127, 27481, 28181, 28631, 28711, 28961, 29059, 29443, 29501, 30169, 31153, 31183, 32213, 32801, 33739, 33797, 33811, 33941, 34283, 35027, 35051, 35729, 35963, 36137, 36251, 36383, 36809, 36943, 37223, 37369, 37511, 37619, 37967, 38281, 38917

Another category of prime involves the average of the prime and the next prime being palindromic. Again 28181 satisfies since:$$ \frac{28181+28183}{2}=28182$$Many such primes are the lesser of a twin prime pair but not all. Primes of this sort constitute OEIS A242387:

A242387: lesser of consecutive primes whose average is a palindromic number.

The members of this sequence up to 40000 have a density of 1.213% compared to all primes and these are (permalink):

3, 5, 7, 97, 109, 281, 359, 389, 409, 509, 631, 653, 691, 743, 827, 857, 907, 937, 967, 1549, 2111, 2767, 4219, 4441, 7001, 9007, 9337, 9661, 10099, 11503, 12919, 13421, 16759, 17569, 21011, 21611, 23831, 26261, 26861, 28181, 29287, 29483, 30497, 31307, 32213, 33029, 33629, 34739, 36353, 37463, 39089

Another category of prime involves the differences between consecutive digits. Some primes have consecutive digits that differ by 6 or 7. An example is 28181 where we see that:$$ 2_{ \, 6} \, 8_{ \, 7} \, 1_{ \, 7} \, 8_{ \, 7} \, 1$$Such primes are few and far between and in the range up 40000, there are only the following:

17, 29, 71, 181, 281, 293, 607, 829, 929, 2939, 3929, 8171, 8293, 9281, 9293, 18181, 28181, 39293

Such primes belong to OEIS A048418:

A048418: primes whose consecutive digits differ by 6 or 7.

Yes another category involves totals of composite numbers between successive primes that are palindromes. 28181 qualifies once again because the next prime is its twin 28183 and the interprime number, 28182, is palindromic. Let's consider another prime, 29587. The next prime is 29599 and the composite numbers between them total 325523, a palindrome. Therefore we include 29587. These primes form OEIS A054266 with a density of only 0.8089% of the primes in the range up to 40000:

A054266: sum of composite numbers between prime \(p\) and nextprime(\(p\)) is palindromic.

The members up to 40000 are (permalink):

2, 3, 5, 109, 193, 281, 509, 661, 827, 857, 1439, 2111, 3433, 3889, 3967, 4549, 6661, 7001, 8467, 10099, 17203, 18583, 21011, 21611, 23831, 24847, 25117, 26261, 26497, 26861, 28181, 29587, 30497, 31307

We see that 28181, my diurnal age today, features in all these different categories of primes. Another category of primes (to which 28181 cannot belong) is to consider primes that only consist of non-prime digits (0, 1, 4, 6, 8 and 9). They do not contain any prime digits (2, 3, 5 or 7). Such primes belong to OEIS A034844 and comprise 5.782% of the primes up to 40000:

A034844: primes with only nonprime decimal digits.

Here are the primes up to 40000 (permalink):

11, 19, 41, 61, 89, 101, 109, 149, 181, 191, 199, 401, 409, 419, 449, 461, 491, 499, 601, 619, 641, 661, 691, 809, 811, 881, 911, 919, 941, 991, 1009, 1019, 1049, 1061, 1069, 1091, 1109, 1181, 1409, 1481, 1489, 1499, 1601, 1609, 1619, 1669, 1699, 1801, 1811, 1861, 1889, 1901, 1949, 1999, 4001, 4019, 4049, 4091, 4099, 4111, 4409, 4441, 4481, 4649, 4691, 4801, 4861, 4889, 4909, 4919, 4969, 4999, 6011, 6089, 6091, 6101, 6199, 6449, 6469, 6481, 6491, 6619, 6661, 6689, 6691, 6841, 6869, 6899, 6911, 6949, 6961, 6991, 8009, 8011, 8069, 8081, 8089, 8101, 8111, 8161, 8191, 8419, 8461, 8609, 8641, 8669, 8681, 8689, 8699, 8819, 8849, 8861, 8941, 8969, 8999, 9001, 9011, 9041, 9049, 9091, 9109, 9161, 9181, 9199, 9419, 9461, 9491, 9601, 9619, 9649, 9661, 9689, 9811, 9901, 9941, 9949, 10009, 10061, 10069, 10091, 10099, 10111, 10141, 10169, 10181, 10499, 10601, 10691, 10861, 10889, 10891, 10909, 10949, 11069, 11119, 11149, 11161, 11411, 11489, 11491, 11681, 11689, 11699, 11801, 11909, 11941, 11969, 11981, 14009, 14011, 14081, 14149, 14401, 14411, 14419, 14449, 14461, 14489, 14669, 14699, 14869, 14891, 14969, 16001, 16061, 16069, 16091, 16111, 16141, 16189, 16411, 16481, 16619, 16649, 16661, 16691, 16699, 16811, 16889, 16901, 16981, 18041, 18049, 18061, 18089, 18119, 18149, 18169, 18181, 18191, 18199, 18401, 18461, 18481, 18661, 18691, 18869, 18899, 18911, 18919, 19001, 19009, 19069, 19081, 19141, 19181, 19441, 19469, 19489, 19609, 19661, 19681, 19699, 19801, 19819, 19841, 19861, 19889, 19891, 19919, 19949, 19961, 19991

Primes beginning with 2 or 3 cannot qualify and so it is only when we reach primes beginning with 4 that membership is possible. The first of these is 40009.

We can flip this and consider only those primes that are comprised of prime digits. These form OEIS A019546:

A019546: primes whose digits are primes; primes having only {2, 3, 5, 7} as digits.

These primes have a density of 2.890% of the primes up to 40000 are they are (permalink):

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557, 5573, 5737, 7237, 7253, 7333, 7523, 7537, 7573, 7577, 7723, 7727, 7753, 7757, 22273, 22277, 22573, 22727, 22777, 23227, 23327, 23333, 23357, 23537, 23557, 23753, 23773, 25237, 25253, 25357, 25373, 25523, 25537, 25577, 25733, 27253, 27277, 27337, 27527, 27733, 27737, 27773, 32233, 32237, 32257, 32323, 32327, 32353, 32377, 32533, 32537, 32573, 33223, 33353, 33377, 33533, 33577, 33757, 33773, 35227, 35257, 35323, 35327, 35353, 35527, 35533, 35537, 35573, 35753, 37223, 37253, 37273, 37277, 37337, 37357, 37537, 37573

Code for Attractors, Vortices and Captives

Herein is an attempt to organise the code that I've gotten Gemini to write for me regarding attractors, vortices and captives.

Firstly, let's start with the ODD(+) and EVEN(-) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 1 and Figure 1). The default range is 0 to 40000.

Table 1: ODD(+) and EVEN(-)

Figure 1: red = attractor, orange = vortex, blue = captive

Secondly, let's continue with the ODD(-) and EVEN(+) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 2 and Figure 2). The default range is 0 to 40000.

Table 2: ODD(-) and EVEN(+)

Figure 2: red = attractor, orange = vortex, blue = captive

Thirdly, let's continue with the PRIME(+) and NON-PRIME(-) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 3 and Figure 3). The default range is 0 to 40000.

Table 3: PRIME(+) and NON-PRIME(-)

Figure 3: red = attractor, orange = vortex, blue = captive

Fourthly, let's continue with the PRIME(-) and NON-PRIME(+) algorithm. Here is a permalink to the code that will generate a list of attractors and vortices in decreasing order of their number of captives. It will also generate a summary and a colour-coded graphical display (see Table 4 and Figure 4). The default range is 0 to 40000.

Table 4: PRIME(-) and NON-PRIME(+)

Figure 4:  red = attractor, orange = vortex, blue = captive

Super Attractors

In my own private terminology, I deem a number an odd-even attractor if its sums of odd digits and even digits are the same. I use the term attractor because numbers that are not attractors are "attracted" to such numbers. For example, let's take the case of 134. Here the sum of the odd numbers is 1 + 3 = 4 and the sum of the even numbers is 4. Thus it is an odd-even attractor. 

Let's take a number like 122 that is not an odd-even attractor. The sum of the even digits (4) exceeds the sum of the odd digits (1).  The difference between odd and even digits is 1 - 4 = -3 and this will be added to the original number to get 119. Now the sum of the odd numbers (11) exceeds that of the non-existent even numbers (0) and this is added to 119 to get 130. Repeating the process we get 134 which is an attractor.

In this system, I've chosen to subtract the sum of the even digits from the sum of the odd digits. This is quite arbitrary and I could have chosen to subtract the sum of the odd digits from the even digits but for odd-even or even-odd attractors this doesn't matter. A similar system can be adopted for prime and non-prime digits. The prime digits are 2, 3, 5 and 7 whereas the non-prime digits are 0, 1, 4, 6, 8 and 9. A number wherein the sum of the prime digits equals that of the non-prime digits is called, in my nomenclature, a prime-non-prime attractor. An example would be 358 where 3 + 5 = 8.

A number that is not a prime-non-prime attractor is 356. Here the sum of prime digits is 8 and the sum of the non-prime digits is 6. We chose to subtract the sum of non-prime digits from the sum of the prime digits to get 2 which we add to 356 to get 358 which is a prime-non-prime attractor.

The question that I was interested in is how many numbers are both odd-even attractors and prime-non-prime attractors? We might term these super attractors. In the range up to 40000, there are 222 such numbers (permalink) and they are:

112, 121, 211, 336, 358, 363, 385, 538, 583, 633, 835, 853, 1012, 1021, 1102, 1120, 1201, 1210, 2011, 2101, 2110, 3036, 3058, 3063, 3085, 3306, 3360, 3445, 3454, 3467, 3476, 3508, 3544, 3580, 3603, 3630, 3647, 3674, 3746, 3764, 3805, 3850, 4345, 4354, 4367, 4376, 4435, 4453, 4534, 4543, 4556, 4565, 4578, 4587, 4637, 4655, 4673, 4736, 4758, 4763, 4785, 4857, 4875, 5038, 5083, 5308, 5344, 5380, 5434, 5443, 5456, 5465, 5478, 5487, 5546, 5564, 5645, 5654, 5667, 5676, 5748, 5766, 5784, 5803, 5830, 5847, 5874, 6033, 6303, 6330, 6347, 6374, 6437, 6455, 6473, 6545, 6554, 6567, 6576, 6657, 6675, 6734, 6743, 6756, 6765, 6778, 6787, 6877, 7346, 7364, 7436, 7458, 7463, 7485, 7548, 7566, 7584, 7634, 7643, 7656, 7665, 7678, 7687, 7768, 7786, 7845, 7854, 7867, 7876, 8035, 8053, 8305, 8350, 8457, 8475, 8503, 8530, 8547, 8574, 8677, 8745, 8754, 8767, 8776, 10012, 10021, 10102, 10120, 10201, 10210, 11002, 11020, 11200, 12001, 12010, 12100, 20011, 20101, 20110, 21001, 21010, 21100, 30036, 30058, 30063, 30085, 30306, 30360, 30445, 30454, 30467, 30476, 30508, 30544, 30580, 30603, 30630, 30647, 30674, 30746, 30764, 30805, 30850, 33006, 33060, 33600, 34045, 34054, 34067, 34076, 34405, 34450, 34504, 34540, 34607, 34670, 34706, 34760, 35008, 35044, 35080, 35404, 35440, 35800, 36003, 36030, 36047, 36074, 36300, 36407, 36470, 36704, 36740, 37046, 37064, 37406, 37460, 37604, 37640, 38005, 38050, 38500

Figure 1 shows the rather uneven distribution of such numbers in the range up to 40000:

Figure 1

All of the numbers greater than 10000 contain the digit 0. Attractors are very much base-specific and thus fall into the realm of recreational mathematics. The big gaps occur between 12100 and 20011 and 21100 and 30036. Numbers that are not attractors of either sort but are close to super attractors do not necessarily end up attracted to the nearest attractor. 

Take 30035 that is next to the super attractor 30036. Here is its prime-non-prime trajectory:

\(30035 \rightarrow 30046 \rightarrow 30039 \rightarrow 30036 \rightarrow 30036\)

While it ends up at the nearby super attractor, the same is not true when subjected to the odd-even trajectory:

\(30035 \rightarrow 30046 \rightarrow 30039 \rightarrow 30054 \rightarrow 30058 \rightarrow 30058\)

It ends up at the more distant super attractor 30058.

Odd and Even Digit Averages

What do the following 98 numbers have in common (permalink)?

102, 123, 147, 234, 258, 306, 345, 369, 456, 567, 678, 789, 1012, 1034, 1056, 1078, 1223, 1245, 1267, 1289, 1447, 1469, 2334, 2356, 2378, 2558, 3036, 3058, 3445, 3467, 3489, 3669, 4556, 4578, 5667, 5689, 6778, 7889, 10004, 10022, 10036, 10112, 10167, 10234, 10356, 10478, 10667, 11125, 11233, 11247, 11459, 11477, 12223, 12278, 12345, 12467, 12589, 12778, 13349, 13457, 14447, 14555, 14569, 14677, 15699, 16779, 20238, 20346, 22236, 22344, 22358, 22588, 23334, 23389, 23456, 23578, 23889, 24568, 25558, 25666, 25788, 30048, 30066, 30336, 30444, 30458, 30566, 33347, 33455, 33469, 33699, 34445, 34567, 34689, 35679, 36669, 36777, 36899

First and foremost, all the numbers have their digits arranged in ascending order. Apart from that however, what else do they have in common? Let's look at the largest member of the sequence: 36899. It has odd digits of 3, 9, 9 and even digits of 6, 8. What are the averages of the odd and even digits? Well, (3 + 9 + 9)/3 = 7 and (6 + 8)/3 = 7 and so the average of the odd and even digits is the same. This is true of all the numbers. The reason that the digits have been arranged in ascending order is that these might be called "root" numbers because any number that is a permutation of the digits of the one of these numbers will have the property that its average of odd and even digits is the same.

In the range up to 40000, there are 1886 numbers that satisfy this equal average criterion. This comprises 4.71% of the range. Here is a permalink that will display these numbers. If we want to apply additional criteria, then this total of 1886 can be reduced significantly. For example, let’s require that the numbers be prime as well as the average of odd and even digits. The first such number would be 1223 which is prime and where the odd average is (1 + 3)/2 = 2 and the even average is (2 + 2)/2 = 2. There are 97 such numbers in the range up to 40000 (permalink):

1223, 1289, 2213, 2819, 2837, 3041, 3221, 3467, 4013, 4637, 4673, 4691, 5689, 5869, 6473, 6491, 7283, 7643, 7687, 7823, 7867, 8219, 8237, 8273, 8291, 8677, 9281, 9461, 10243, 11251, 12043, 12511, 12589, 14767, 15121, 15289, 15649, 16477, 16747, 17467, 17827, 20143, 20341, 20431, 21313, 21589, 21787, 21859, 22123, 23041, 23131, 23311, 23857, 23893, 24103, 25111, 25189, 25819, 25873, 25981, 27583, 27817, 28393, 28537, 28573, 28591, 28753, 28771, 28933, 29383, 29581, 29833, 29851, 30241, 31123, 31231, 31321, 32401, 32587, 32839, 32983, 33211, 33289, 33469, 33829, 34369, 34693, 34963, 36457, 36493, 36899, 36943, 38239, 38329, 38699, 38923, 39869

Neither of these two sequences is registered in the OEIS and I certainly won't be proposing them to the cretins who oversee it. However, the sequences are in my own private database. Here is the link. The averages themselves do not need to be integers. For example 3863503 has an odd average of (3 + 3 + 5 + 3)/3 = 14/3 and an even average of (8 + 6 + 0)/3 = 14/3. However, for all numbers in the range up to 40000, the averages are all integers.

Of course, we need not consider odd and even digits. We could consider the average of prime (2, 3, 5 and 7) and non-prime (0, 1, 4, 6, 8, 9) digits. An example of a number in which the average of the prime and non-prime digits is the same would be 39760 where (3 + 7)/2 = 5 and (9 + 6 + 0)/3 = 5. Here is a permalink to generate these numbers.

A Semiprime Rara Avis

A rara avis or rare bird is someone or something that is rare and this type of semiprime is indeed rare. It came to my attention recently when I looked at a number associated with my diurnal age that, at the time, was 27331 days old. This number is a semiprime and factorises to:$$27331= \underbrace{151}_{ \text{19th 4k+3 prime}} \times \underbrace{181}_{ \text{19th 4k+1 prime}}$$Numbers like 27331 form OEIS A048630 (permalink) and remember that \(4k-1 \equiv 4k+3\):

A048630

\(n\)-th 4\(k\)+1 prime times \(n\)-th 4\(k\)-1 prime.

The first 22 of these semiprimes are as follows:

• 1 --> 15 = 3 x 5

• 2 --> 91 = 7 x 13

• 3 --> 187 = 11 x 17

• 4 --> 551 = 19 x 29

• 5 --> 851 = 23 x 37

• 6 --> 1271 = 31 x 41

• 7 --> 2279 = 43 x 53

• 8 --> 2867 = 47 x 61

• 9 --> 4307 = 59 x 73

• 10 --> 5963 = 67 x 89

• 11 --> 6887 = 71 x 97

• 12 --> 7979 = 79 x 101

• 13 --> 9047 = 83 x 109

• 14 --> 11639 = 103 x 113

• 15 --> 14659 = 107 x 137

• 16 --> 18923 = 127 x 149

• 17 --> 20567 = 131 x 157

• 18 --> 24047 = 139 x 173

• 19 --> 27331 = 151 x 181

• 20 --> 31459 = 163 x 193

• 21 --> 32899 = 167 x 197

• 22 --> 40991 = 179 x 229

As can be seen, my next "experience" of such a semiprime will come when I am 31459 days old. This will occur on Monday, May 21st 2035, not long after my 86th birthday if I make it that far. So in terms of life experiences, marked by the passing of the days, most individuals will be lucky if they see 21 of them. In my case the 21st falls on Saturday, April 30th 2039, three weeks after my 90th birthday. Nobody will experience the 22nd such semiprime!

Working with the \(n\)-th 4\(k\)+1 prime and the \(n\)-th 4\(k\)-1 prime, we could generate other sequences such as for example the average of the \(n\)-th 4\(k\)+1 prime and the \(n\)-th 4\(k\)-1 prime. This sequence is not listed in the OEIS but it begins with 4, 10, 14, 24, 30, 36, 48, 54, 66, 78, 84, 90, 96, 108, 122, 138, 144, 156, 166, 178, 182, 204 (permalink). These sorts of sequences could be termed "hybrid" because they involve the combination of two related but separate sequences.

We could take the sum of the \(n\)-th 4\(k\)+1 prime and the \(n\)-th 4\(k\)-1 prime and add 1 to generate a sequence of odd numbers: 9, 21, 29, 49, 61, 73, 97, 109, 133, 157, 169, 181, 193, 217, 245, 277, 289, 313, 333, 357, 365, 409 etc. (permalink). Again, this is not listed in the OEIS. All sorts of combinations are possible between these and other sequences, the results of which might be of mathematical interest but most fall into the realm of recreational mathematics.

In a recent post, Yet Another Type of Prime on January 19th 2024, I examined hybrid sequences arising from the sum of the \(n\)-th prime and the \(n\)-th composite number as well as the sum of the  \(n\)-th prime and the \(n\)-th non-prime number. I went on to look at subsequences involving the primes arising within these summed sequences.  

Yet Another Type of Prime

Before we delve into the world of prime numbers, we firstly need to consider a special type of number or rather two types of numbers that are similar and yet distinct:

• numbers that are the sum of the \(n\)-th prime and the \(n\)-th non-prime number
• numbers that are the sum of the \(n\)-th prime and the \(n\)-th composite number

The first non-prime numbers is 1 while the first composite number is 4. All other non-prime and composite numbers are the same but the different starting points produce two different sequences, one beginning with 1 + 2 = 3 and the other beginning with 4 + 2 = 6:

• 3, 7, 11, 15, 20, 23, 29, 33, 38, 45, 49, 57, 62, 65, 71, 78, 85, 88, 95, ... OEIS  A064799
• 6, 9, 13, 16, 21, 25, 31, 34, 39, 47, 51, 58, 63, 67, 72, 79, 86, 89, 97, ... OEIS  A064799

The primes we are interested in are those primes to be found in these two sequences. The first is OEIS A097452:

A097452

Primes of the form prime(\(n\)) + nonprime(\(n\)) for some \(n\).

Up to 40,000, there are 484 of these types of primes (see Bespoken for Sequences entry):

3, 7, 11, 23, 29, 71, 101, 139, 151, 157, 199, 229, 239, 251, 263, 311, 347, 367, 401, 443, 479, 547, 601, 653, 673, 691, 709, 853, 977, 991, 1013, 1051, 1087, 1181, 1237, 1291, 1327, 1451, 1487, 1579, 1637, 1693, 1721, 1753, 1777, 1861, 1877, 1913, 1951, 2029, 2087, 2161, 2237, 2251, 2297, 2351, 2381, 2543, 2557, 2657, 2683, 2767, 2777, 2791, 2897, 3011, 3079, 3121, 3169, 3209, 3221, 3299, 3413, 3461, 3499, 3571, 3623, 3631, 3719, 3739, 3779, 3823, 3919, 4021, 4129, 4231, 4253, 4297, 4327, 4409, 4421, 4483, 4507, 4547, 4567, 4583, 4637, 4673, 4733, 4801, 4937, 4951, 4973, 4987, 5087, 5399, 5743, 5807, 5813, 5821, 5923, 6047, 6067, 6269, 6277, 6343, 6353, 6451, 6551, 6733, 6997, 7019, 7027, 7457, 7481, 7589, 7829, 7841, 7877, 8111, 8297, 8317, 8539, 8627, 8647, 8681, 8693, 8707, 8737, 8747, 8929, 8999, 9013, 9067, 9293, 9319, 9337, 9397, 9419, 9439, 9473, 9887, 9949, 10009, 10037, 10067, 10301, 10333, 10343, 10391, 10487, 10589, 10663, 10691, 10853, 10861, 10993, 11057, 11117, 11177, 11197, 11213, 11239, 11317, 11351, 11527, 11681, 11701, 11867, 11971, 12011, 12143, 12281, 12527, 12589, 12899, 12983, 13033, 13367, 13513, 13523, 13553, 13619, 13627, 13691, 13831, 13931, 13999, 14029, 14153, 14177, 14251, 14327, 14537, 14557, 14723, 14783, 14867, 14879, 14939, 14951, 15107, 15173, 15193, 15199, 15289, 15473, 15601, 15647, 15733, 15907, 16067, 16111, 16223, 16301, 16567, 16573, 16693, 16741, 16747, 16763, 16811, 16931, 16987, 17033, 17099, 17123, 17327, 17389, 17419, 17471, 17509, 17627, 17707, 17791, 17839, 17939, 17959, 17989, 18191, 18397, 18691, 18719, 18803, 18919, 18959, 18973, 19013, 19219, 19423, 19441, 19559, 19681, 19751, 19861, 19919, 20101, 20143, 20287, 20333, 20509, 20681, 20749, 20807, 20903, 20939, 21179, 21283, 21317, 21433, 21499, 21529, 21563, 21757, 21773, 21851, 22003, 22039, 22063, 22123, 22307, 22381, 22469, 22573, 22613, 22697, 22777, 22807, 22853, 23011, 23027, 23063, 23143, 23189, 23311, 23447, 23497, 23531, 23603, 23627, 23801, 23833, 23899, 23981, 24001, 24121, 24421, 24439, 24623, 24631, 24659, 24923, 24953, 25057, 25097, 25229, 25309, 25589, 25673, 25763, 25951, 25997, 26029, 26041, 26111, 26209, 26227, 26297, 26321, 26641, 26903, 26927, 26993, 27011, 27091, 27427, 27509, 27697, 27809, 27851, 27953, 28051, 28123, 28297, 28351, 28547, 28571, 28591, 28597, 28687, 28837, 29063, 29327, 29333, 29363, 29389, 29437, 29581, 29629, 29683, 29761, 29863, 30047, 30091, 30109, 30307, 30391, 30509, 30517, 30529, 30557, 30661, 30911, 30971, 31081, 31159, 31259, 31267, 31387, 31517, 31573, 31643, 32003, 32063, 32117, 32143, 32159, 32183, 32251, 32401, 32579, 32587, 32611, 32801, 33013, 33037, 33053, 33091, 33161, 33403, 33413, 33461, 33487, 33619, 33641, 33679, 33751, 33923, 33997, 34019, 34039, 34157, 34217, 34367, 34487, 34511, 34613, 34687, 34913, 35089, 35153, 35281, 35327, 35401, 35447, 35617, 35759, 35831, 36011, 36037, 36083, 36187, 36263, 36313, 36343, 36467, 36541, 36587, 36683, 36787, 36821, 36913, 36923, 36997, 37117, 37181, 37337, 37483, 37537, 37643, 37663, 37691, 37799, 37861, 37897, 37957, 37991, 38351, 38453, 38501, 38543, 38569, 38639, 38821, 38873, 39019, 39113, 39157, 39229, 39607, 39631, 39821, 39883, 39953

An example is 27427 = 3146 + 24281 which is the sum of the 2700-th non-prime and prime numbers. The second sequence of primes is OEIS A111489:

A111489

Primes of the form prime(\(n\)) + composite(\(n\)) for some \(n\).

Up to 40,000, there are 447 such primes (see Bespoken for Sequences entry):

13, 31, 47, 67, 79, 89, 97, 103, 113, 149, 173, 179, 211, 223, 241, 277, 313, 349, 359, 379, 449, 457, 487, 503, 509, 631, 743, 769, 797, 809, 887, 937, 967, 1009, 1049, 1109, 1123, 1213, 1231, 1277, 1289, 1319, 1409, 1429, 1453, 1471, 1489, 1543, 1571, 1663, 1709, 1747, 1789, 1801, 1879, 1999, 2081, 2137, 2377, 2383, 2399, 2411, 2459, 2531, 2539, 2617, 2633, 2687, 2693, 2819, 2843, 2927, 2999, 3023, 3089, 3203, 3301, 3347, 3449, 3463, 3529, 3557, 3733, 3821, 3877, 3907, 4001, 4057, 4111, 4133, 4261, 4363, 4423, 4451, 4519, 4549, 4691, 4759, 4789, 4877, 4903, 4999, 5081, 5099, 5119, 5413, 5441, 5449, 5563, 5651, 5669, 5737, 5779, 5801, 5839, 5869, 5897, 5903, 5927, 6073, 6247, 6271, 6299, 6311, 6359, 6379, 6473, 6571, 6607, 6619, 6653, 6719, 6763, 6791, 6977, 7129, 7243, 7253, 7321, 7477, 7583, 7591, 7691, 7741, 7883, 7933, 7949, 8171, 8291, 8329, 8429, 8521, 8731, 8761, 8839, 8969, 9007, 9041, 9103, 9137, 9151, 9281, 9311, 9371, 9421, 9631, 9689, 9767, 9967, 9973, 10039, 10069, 10093, 10181, 10211, 10271, 10337, 10369, 10457, 10781, 10799, 10831, 10889, 10957, 11171, 11257, 11353, 11399, 11549, 11587, 11617, 11807, 11827, 12041, 12239, 12329, 12401, 12437, 12517, 12647, 12689, 12763, 12781, 12853, 12893, 12923, 13001, 13109, 13127, 13151, 13177, 13241, 13339, 13451, 13751, 13873, 14143, 14197, 14243, 14321, 14369, 14423, 14437, 14519, 14543, 14629, 14639, 14653, 14831, 14843, 14887, 15013, 15161, 15241, 15277, 15349, 15377, 15641, 15671, 15791, 15877, 15913, 15959, 16139, 16447, 16561, 16649, 16661, 16879, 16901, 16921, 16981, 17021, 17167, 17291, 17321, 17393, 17489, 17623, 17789, 17981, 18089, 18181, 18199, 18251, 18287, 18307, 18401, 18439, 18593, 18671, 18797, 18859, 18913, 19001, 19213, 19289, 19333, 19391, 19457, 19489, 19753, 20129, 20269, 20323, 20411, 20479, 20593, 20627, 20639, 20747, 20809, 21013, 21089, 21193, 21617, 21683, 21767, 21787, 21991, 22013, 22091, 22157, 22369, 22397, 22481, 22621, 22637, 22669, 22679, 23029, 23053, 23333, 23629, 23677, 23719, 23743, 23831, 24113, 24593, 24733, 24917, 24989, 25189, 25321, 25423, 25453, 25609, 25639, 25759, 25841, 25933, 26099, 26309, 26371, 26561, 26591, 26627, 26699, 26821, 26891, 26953, 26987, 27239, 27253, 27271, 27337, 27397, 27487, 27631, 27883, 28019, 28081, 28387, 28559, 28573, 28627, 28817, 29059, 29179, 29251, 29527, 29717, 29803, 29819, 30071, 30137, 30203, 30389, 30677, 30707, 30841, 30893, 30937, 31039, 31051, 31121, 31153, 31181, 31247, 31271, 31319, 31327, 31333, 31357, 31489, 31511, 31583, 31721, 31859, 31873, 31907, 32257, 32321, 32369, 32467, 32779, 33071, 33347, 33427, 33469, 33587, 33713, 33871, 34127, 34267, 34313, 34501, 34519, 34607, 34781, 34981, 35083, 35171, 35323, 35491, 35531, 35993, 36061, 36161, 36277, 36583, 36629, 36637, 36697, 36761, 36791, 36877, 36929, 37223, 37339, 37423, 37447, 37489, 37907, 38153, 38449, 38693, 38723, 38959, 39161, 39191, 39503, 39619, 39667, 39727, 39887

An example is 27337 = 3140 + 24197 which is the sum of the 2693-th composite and prime numbers. One could take things a step further for these two sequences and consider only members of the sequence for \(n\) prime. Thus only the 2nd, 3rd, 5th, 7th, 11th and so on sequence members would appear. Consider the following sequence (not in the OEIS) and accompanying table.

Primes of the form prime(\(n\)) + composite(\(n\)) for \(n\) prime.