Erdős–Selfridge classification of primes

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The Erdős–Selfridge classification of primes classifies prime numbers according to their neighbors.

Paul Erdős and John Selfridge classified primes

p

as:

• A prime

  p

  is in class 1+ if

  ( p  +  1)

  ’s greatest prime factor is 2 or 3;

• If

  ( p + 1)

  has other prime factors,

  p

  ’s class is one more than the largest class of its prime factors.

• A prime

  p

  is in class 1− if

  ( p  −  1)

  ’s largest prime factor is 2 or 3;

• If

  ( p  −  1)

  has other prime factors,

  p

  ’s class is one less than the smallest class of its prime factors.

All Mersenne primes (primes of the form

2 p  −  1

,

p

prime) are in class 1+. All Fermat primes (primes of the form

2 2 n  +  1, n   ≥   0

) are in class 1−.

Sequences

A?????? “Class+” number of prime(

n

) − “Class−” number of prime(

n

). (A126433(

n

) − A126805(

n

).)

{0, 0, 0, 0, −1, 1, 0, 1, −2, 0, −1, 2, 0, 0, −3, −1, −1, 0, −1, −1, 3, 0, −1, −1, 1, 0, 1, −2, 1, 1, −1, 0, 0, −2, 0, 1, 1, 2, −2, 0, −2, 1, −1, 2, 0, −1, 0, 0, 0, 0, 0, 0, 0, 0, 2, −1, −2, 0, −1, 0, −3, 0, 0, 0, 2, −1, 0, 1, −1, ...}

The partial sums of the above sequence seem to reveal a negative bias... Does this effect persist or not?

A178382 Primes that are in classes 

k +

and 

k −

for some

k

in the Erdős-Selfridge classification of primes. (Prime(

k

) such that A126433(

k

) − A126805(

k

) = 0.)

{2, 3, 5, 7, 17, 29, 41, 43, 61, 79, 101, 131, 137, 149, 173, 197, 211, 223, 227, 229, 233, 239, 241, 251, 271, 281, 293, 307, 311, 331, 353, 397, 439, 449, 463, 523, 569, 593, 607, 641, 683, 691, 727, ...}

A?????? Prime(

k

) such that A126433(

k

) − A126805(

k

) < 0.

{11, 23, 31, 47, 53, 59, 67, 71, 83, 89, 107, 127, 139, 167, 179, 191, 199, 263, 269, 277, 283, 317, 347, ...}

A?????? Prime(

k

) such that A126433(

k

) − A126805(

k

) > 0.

{13, 19, 37, 73, 97, 103, 109, 113, 151, 157, 163, 181, 193, 257, 313, 337, ...}

Class n + primes

A126433 “Class+” (or “class-plus”) number of prime(

n

) according to the Erdős-Selfridge classification of primes.

{1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 1, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 4, 2, 3, 3, 3, 2, 3, 2, 2, 2, 3, 1, 3, 3, 3, ...}

A005113

a (n)

is the least prime of class

n, n   ≥   1,

(sometimes written

n +

) according to the Erdős–Selfridge classification of primes.

{2, 13, 37, 73, 1021, 2917, 15013, 49681, 532801, 1065601, 8524807, 68198461, 545587687, 1704961513, 23869461181, 288310406533, ...}

A005105 Class 1+ primes (or Pierpont primes of the second kind): primes of the form

2 i 3   j  −  1

with

i, j   ≥   0

.

{2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8191, 8747, 13121, 15551, 23327, 27647, 62207, 73727, 131071, ...}

A005106 Class 2+ primes.

{13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, 109, 131, 137, 139, 149, 167, 179, 197, 199, 211, 223, 229, 239, 241, 251, 263, 269, 271, 281, 283, 293, 307, 317, 349, ...}

A005107 Class 3+ primes.

{37, 103, 113, 151, 157, 163, 173, 181, 193, 227, 233, 257, 277, 311, 331, 337, 347, 353, 379, 389, 397, 401, 409, 421, 457, 463, 467, 487, 491, 521, 523, 541, 547, 569, 571, ...}

A005108 Class 4+ primes.

{73, 313, 443, 617, 661, 673, 677, 691, 739, 757, 823, 887, 907, 941, 977, 1093, 1109, 1129, 1201, 1213, 1303, 1361, 1447, 1453, 1543, 1553, 1621, 1627, 1657, 1753, 1811, ...}

A081633 Class 5+ primes.

{1021, 1321, 1381, 1459, 1877, 2467, 2503, 2657, 2707, 3253, 3313, 3547, 3701, 3733, 3907, 4561, 4817, 4937, 5441, 5443, 5527, 5693, 5839, 5861, 6037, 6131, 6211, 6217, ...}

A081634 Class 6+ primes.

{2917, 4933, 5413, 7507, 8167, 8753, 10567, 10627, 11047, 11261, 11677, 12073, 12251, 12421, 12433, 12553, 12721, 14293, 15017, 17041, 18181, 18493, 19267, 19333, ...}

A081635 Class 7+ primes.

{15013, 16333, 22093, 24841, 43321, 49003, 52517, 54721, 62533, 63761, 69061, 69073, 70061, 74597, 75781, 75793, 75913, 82561, 83233, 84673, 87433, 87509, 88793, ...}

A081636 Class 8+ primes.

{49681, 109441, 120103, 151561, 198733, 210193, 246241, 255043, 266401, 280243, 295873, 326659, 326701, 347773, 355171, 360421, 368881, 397633, 397673, 423001, ...}

A081637 Class 9+ primes.

{532801, 710341, 720617, 1212487, 1261157, 1372081, 1457293, 1490429, 1532173, 1657801, 1788547, 1789093, 1809601, 1829293, 1887877, 1944181, 1960141, 1997587, ...}

A081638 Class 10+ primes.

{1065601, 2424973, 5114881, 7222561, 8124481, 8524091, 8647411, 8650321, 9190681, 9287521, 9590417, 10617601, 10929817, 11996161, 12349093, 12508081, 12786181, ...}

A081639 Class 11+ primes.

{8524807, 18381361, 18575041, 19180817, 21312019, 31984321, 34099231, 40357021, 44206633, 44839273, 48499459, 51148847, 51444961, 51884467, 54144121, 57129613, ...}

A084071 Class 12+ primes.

{68198461, 115084901, 138358573, 156811273, 213397621, 220576331, 234432217, 260050573, 282261961, 290996753, 330864497, 353653063, 371500819, 383616341, ...}

A090468 Class 13+ primes.

{545587687, 852480757, 1048438561, 1150849009, 1323457987, 1745980517, 1756123441, 1785398401, 1798736161, 1892507347, 1937020021, 2002155601, 2136716521, ...}

A129474 Class 14+ primes.

{1704961513, 7281416041, 7638227617, 9462536833, 11934730597, 13237911481, 13282423003, 13522629793, 13942983841, 14185279861, 16029089501, 16221987853, ...}

A129475 Class 15+ primes.

{23869461181, 39279010921, 45608421601, 58345550881, 64788537493, 79681330633, 83807064853, 86315197987, 91658731403, 97331927117, 102581556673, ...}

A?????? Class 16+ primes.

{?, ...}

Class n − primes

A126805 “Class−” (or “class-minus”) number of prime(

n

) according to the Erdős-Selfridge classification of primes.

{1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 4, 2, 3, 2, 3, 2, 1, 2, 3, 3, 1, 2, 2, 3, 1, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 3, 4, 2, 2, 1, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 1, 3, 4, 2, 4, 2, 5, 2, 2, 3, 2, 3, 3, 2, 4, 3, 3, 5, 3, 3, 2, 3, 2, 3, 2, 2, ...}

A056637

a (n)

is the least prime of class

n −, n   ≥   1,

according to the Erdős–Selfridge classification of primes.

{2, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107, 1432349099, 22111003847, 110874748763, ...}

A005109 Class 1− primes (or Pierpont primes): primes of the form

2 i 3   j  +  1

with

i, j   ≥   0

.

{2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, ...}

A005110 Class 2− primes.

{11, 29, 31, 41, 43, 53, 61, 71, 79, 101, 103, 113, 127, 131, 137, 149, 151, 157, 181, 191, 197, 211, 223, 229, 239, 241, 251, 271, 281, 293, 307, 313, 337, 379, 389, 401, 409, ...}

A005111 Class 3− primes.

{23, 59, 67, 83, 89, 107, 173, 199, 227, 233, 263, 311, 317, 331, 349, 353, 367, 373, 383, 397, 419, 431, 463, 479, 503, 509, 523, 563, 569, 587, 607, 617, 619, 661, 683, 727, ...}

A005112 Class 4− primes.

{47, 139, 167, 179, 269, 277, 347, 461, 467, 499, 599, 643, 691, 709, 797, 827, 829, 839, 857, 863, 967, 997, 1013, 1019, 1039, 1063, 1069, 1151, 1163, 1181, 1289, 1367, ...}

A081424 Class 5− primes.

{283, 359, 557, 659, 941, 1109, 1129, 1223, 1433, 1663, 1669, 1693, 1787, 1997, 2027, 2039, 2069, 2083, 2153, 2339, 2351, 2503, 2539, 2579, 2633, 2767, 2777, 2803, 2837, ...}

A081425 Class 6− primes.

{719, 1319, 1699, 2447, 3343, 4079, 4139, 4457, 4517, 4679, 4703, 5273, 5647, 6653, 6793, 7523, 7529, 7559, 8599, 9227, 9587, 9623, 9839, 10159, 10343, 10723, ...}

A081426 Class 7− primes.

{1439, 8629, 10067, 14683, 17257, 19577, 20389, 22643, 23743, 27103, 28219, 29399, 31657, 32633, 33107, 33113, 33863, 34259, 34513, 35951, 36137, 36887, 37379, ...}

A081427 Class 8− primes.

{2879, 20147, 25903, 34537, 46049, 58733, 63317, 65267, 69029, 69073, 74759, 80537, 86291, 86341, 103549, 106487, 108413, 112877, 120877, 131687, 135859, ...}

A081428 Class 9− primes.

{34549, 86371, 103613, 120919, 138059, 149519, 172583, 172741, 224563, 276293, 282059, 282143, 293659, 299417, 316691, 352399, 368513, 379903, 397303, 403061, ...}

A081429 Class 10− primes.

{138197, 207227, 621679, 621883, 633383, 760079, 829177, 863711, 898253, 978863, 1035499, 1036471, 1209191, 1451059, 1566179, 1658309, 1658353, 1761407, ...}

A081430 Class 11− primes.

{1266767, 1520159, 2486717, 3316619, 4144541, 4512947, 4836779, 5389519, 5638379, 6218827, 6448979, 6633457, 6771419, 6907247, 7460149, 7462639, 7600597, ...}

A081640 Class 12− primes.

{14920303, 18224639, 24867247, 26532953, 34548443, 38003011, 39800743, 41319599, 41443483, 45604771, 46432667, 47247763, 49734341, 49734493, 49749439, ...}

A081641 Class 13− primes.

{36449279, 53065907, 59681213, 69096887, 132756479, 135388367, 164255999, 179043637, 188991053, 207290663, 241560239, 279709259, 309550999, 364492781, ...}

A129248 Class 14− primes.

{377982107, 437391349, 716174549, 742922699, 1385934359, 1603768277, 1780127639, 1790436371, 1895437139, 1968261067, 2066951933, 2109424013, ...}

A129249 Class 15− primes.

{1432349099, 1749565397, 2771868719, 3790874279, 5288908679, 5804138567, 6273146879, 8123301983, 8594094589, 11055501923, 11809566403, 11914176299, ...}

A129250 Class 16− primes.

{22111003847, 25782283783, 34824831403, 42970472971, 44905511759, 45490491349, 52486961911, 54560052479, 55437374381, 65803884467, 66333011539, ...}