Friday, 19 January 2024

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. 

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