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# Decomposition into weight * level + jump/sequences

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# Sequences related to natural numbers

Lpf(n): least prime dividing n (with a(1)=1). (Cf. A020639(n), n ≥ 1) (referred to as the weight of natural numbers)

{1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, ...}

Largest proper divisor of n (with a(1)=1). (Cf. A032742(n), n ≥ 1) (referred to as the level of natural numbers)

{1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 5, 13, 9, 14, 1, 15, 1, 16, 11, 17, 7, 18, 1, 19, 13, 20, 1, 21, 1, 22, 15, 23, 1, 24, 7, 25, 17, 26, 1, 27, 11, 28, 19, 29, 1, 30, 1, 31, 21, 32, 13, 33, 1, ...}

# Sequences related to primes

The prime numbers. (Cf. A000040(n), n ≥ 1)

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, ...}

Differences between consecutive primes. (Cf. A001223(n), n ≥ 1) (referred to as the gap or jump of primes)

{1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, ...}

a(n) = smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists. (Cf. A117078(n), n ≥ 1) (referred to as the weight of prime numbers)

{0, 0, 3, 0, 3, 9, 3, 5, 17, 3, 25, 11, 3, 13, 41, 47, 3, 11, 7, 3, 67, 5, 7, 9, 31, 3, 9, 3, 5, 33, 41, 25, 3, 43, 3, 29, 151, 53, 7, 167, 3, 19, 3, 7, 3, 17, 199, 73, 3, 5, 227, 3, 11, 7, 251, 257, 3, 53, 7, 3, 13, 31, 101, 3, ...}

a(n) = A118534(n) / A117078(n) unless A117078(n) = 0 in which case a(n) = 0. (Cf. A117563(n), n ≥ 1) (referred to as the level of prime numbers)

{0, 0, 1, 0, 3, 1, 5, 3, 1, 9, 1, 3, 13, 3, 1, 1, 19, 5, 9, 23, 1, 15, 11, 9, 3, 33, 11, 35, 21, 3, 3, 5, 45, 3, 49, 5, 1, 3, 23, 1, 59, 9, 63, 27, 65, 11, 1, 3, 75, 45, 1, 79, 21, 35, 1, 1, 89, 5, 39, 93, 21, 9, 3, 103, 3, 3, 25, 3, ...}

a(n) = largest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists. (Cf. A118534(n), n ≥ 1) (referred to as l(n) of prime numbers)

{0, 0, 3, 0, 9, 9, 15, 15, 17, 27, 25, 33, 39, 39, 41, 47, 57, 55, 63, 69, 67, 75, 77, 81, 93, 99, 99, 105, 105, 99, 123, 125, 135, 129, 147, 145, 151, 159, 161, 167, 177, 171, 189, 189, 195, 187, 199, 219, 225, ...}

a(n) = number of k's such that prime(n+1) = prime(n) + (prime(n) mod k). (Cf. A118123(n), n ≥ 1)

{0, 0, 1, 0, 2, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 1, 3, 2, 4, 3, 1, 4, 3, 3, 2, 5, 4, 7, 6, 2, 2, 2, 7, 2, 5, 2, 1, 2, 3, 1, 3, 3, 7, 6, 7, 2, 1, 2, 8, 7, 1, 3, 5, 4, 1, 1, 3, 2, 6, 5, 5, 3, 2, 3, 2, 2, 4, 2, 7, 6, 1, 6, 2, 1, 6, 3, 2, 2, 2, 5, 3, 2, ...}

a(n) = smallest k such that prime(n+2) = prime(n) + (prime(n) mod k), or 0 if no such k exists. (Cf. A133346(n), n ≥ 1)

{0, 0, 0, 0, 0, 7, 11, 0, 15, 21, 21, 31, 7, 11, 35, 9, 17, 17, 61, 9, 21, 23, 23, 77, 7, 19, 97, 101, 91, 19, 13, 41, 25, 127, 47, 139, 21, 17, 31, 11, 167, 13, 37, 11, 61, 25, 39, 7, 13, 73, 9, 227, 25, 239, 35, 15, 9, ...}

a(n) = smallest k such that prime(n+3) = prime(n) + (prime(n) mod k), or 0 if no such k exists. (Cf. A133347(n), n ≥ 1)

{0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 19, 27, 29, 27, 33, 39, 47, 49, 55, 59, 19, 61, 65, 15, 29, 31, 31, 29, 29, 89, 23, 113, 41, 121, 15, 27, 47, 21, 17, 31, 15, 33, 61, 25, 57, 57, 193, 71, 43, 31, 43, 221, 73, 233, 27, 83, ...}

Numbers of prime factors of k, k defined in A117078. (Cf. A106752(n), n ≥ 1)

{0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

Numbers of prime factors of l, l defined in A118534. (Cf. A118144 (n), n ≥ 1)

{0, 0, 1, 0, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 1, 2, 2, 3, 2, 1, 3, 2, 4, 2, 3, 3, 3, 3, 3, 2, 3, 4, 2, 3, 2, 1, 2, 2, 1, 2, 3, 4, 4, 3, 2, 1, 2, 4, 4, 1, 2, ...}

Primes for which A117078(n) is equal to A117563(n) and A117563(n) is different from 0. (Cf. A121155(n), n ≥ 1)

{11, 89, 367, 3727, 5059, 7927, 38821, 94261, 160807, 727621, 908221, 942847, 1274671, 1985287, 2042059, 2105407, 2411821, 4068301, 4464781, 4748077, 5004193, 5331511, 5678713, 5755219, ...}

## Sequences related to primes classified by weight

Primes classified by weight. (Cf. A162175(n), n ≥ 1)

{11, 17, 29, 41, 59, 67, 71, 79, 83, 89, 101, 103, 107, 109, 137, 149, 167, 179, 191, 193, 197, 227, 229, 239, 241, 251, 269, 277, 281, 283, 311, 331, 347, 349, 359, 367, 379, 383, 409, 419, 431, 433, 439, ...}

Lesser of twin primes - {3} : primes for which the weight as defined in A117078 is 3. (Cf. A001359(n), n ≥ 2)

{5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, ...}

Primes for which the weight as defined in A117078 is 5. (Cf. A074822(n), n ≥ 1)

{19, 79, 109, 229, 349, 379, 439, 499, 739, 769, 859, 1009, 1279, 1429, 1489, 1549, 1579, 1609, 1999, 2239, 2269, 2389, 2539, 2659, 2689, 2749, 3019, 3079, 3319, 3529, 3919, 4129, 4519, 4639, 4729, ...}

Primes for which the weight as defined in A117078 is 7. (Cf. A118741(n), n ≥ 1)

{67, 83, 167, 193, 251, 277, 433, 487, 503, 587, 601, 613, 727, 823, 907, 1063, 1217, 1231, 1553, 1663, 1777, 1861, 1873, 1973, 1987, 2083, 2281, 2293, 2351, 2377, 2393, 2797, 2897, 3217, 3343, 3541, ...}

Primes for which the weight as defined in A117078 is 7 and the gap or jump as defined in A001223 is 4. (Cf. A119593(n), n ≥ 1)

{67, 193, 277, 487, 613, 823, 907, 1663, 1873, 2083, 2293, 2377, 2797, 3217, 3343, 3847, 4813, 5233, 5527, 5653, 5737, 6577, 6997, 7207, 7753, 8677, 8803, 9433, 11113, 11617, ...}

Primes for which the weight as defined in A117078 is 7 and the gap or jump as defined in A001223 is 6. (Cf. A118359(n), n ≥ 1)

{83, 167, 251, 433, 503, 587, 601, 727, 1063, 1217, 1231, 1553, 1777, 1861, 1973, 1987, 2281, 2351, 2393, 2897, 3541, 4073, 4283, 4451, 4507, 4591, 4871, 5081, 5431, 5557, 5641, 5683, ...}

Primes for which the weight as defined in A117078 is 9 and the gap or jump as defined in A001223 is 4. (Cf. A119594(n), n ≥ 1)

{13, 103, 463, 643, 877, 967, 1093, 1597, 1867, 1993, 2137, 2857, 3037, 3163, 3253, 3613, 3793, 4153, 4513, 4783, 5413, 5503, 5647, 6007, 6043, 6547, 6907, 7537, 7573, 7933, 8167, 8293, 9157, 9337, ...}

Primes for which the weight as defined in A117078 is 9 and the gap or jump as defined in A001223 is 8. (Cf. A118922(n), n ≥ 1)

{89, 359, 449, 683, 701, 719, 1439, 1979, 2213, 2609, 2663, 2699, 2843, 2879, 3041, 3221, 3491, 4751, 5399, 5813, 6029, 6389, 6983, 7019, 7919, 8171, 8369, 8513, 9539, 10151, 10169, 10259, 10313, ...}

Primes for which the weight as defined in A117078 is 11 and the gap or jump as defined in A001223 is 6. (Cf. A119597(n), n ≥ 1)

{61, 677, 941, 1117, 1601, 2063, 2371, 3691, 3911, 4021, 5297, 5407, 6067, 6353, 6991, 7541, 7717, 8311, 8641, 8663, 9103, 9851, 10973, 11897, 12491, 12953, 13591, 13613, 13723, 14537, 15131, 15263, ...}

Primes for which the weight as defined in A117078 is 11 and the gap or jump as defined in A001223 is 10. (Cf. A119596(n), n ≥ 1)

{241, 1627, 2089, 4201, 4663, 4861, 5323, 6247, 6379, 6709, 8821, 9283, 9679, 10141, 12253, 12517, 12781, 13441, 15091, 15289, 15619, 17599, 17929, 19249, 19447, 19843, 21757, 23539, 26839, ...}

Primes for which the weight as defined in A117078 is 15 and the gap or jump as defined in A001223 is 8. (Cf. A119595(n), n ≥ 1)

{743, 1193, 1523, 1733, 2003, 2243, 2273, 3623, 4583, 4943, 5573, 5693, 6143, 6203, 6473, 7673, 8573, 8933, 9803, 10103, 11243, 11813, 12413, 12503, 13163, 14423, 14843, 15053, 15233, 15383, 16103, ...}

Primes for which the weight as defined in A117078 is 15 and the gap or jump as defined in A001223 is 14. (Cf. A118380(n), n ≥ 1)

{839, 1409, 2039, 2819, 2939, 3779, 4139, 4889, 5309, 5669, 5939, 6719, 8039, 8609, 10739, 11369, 11909, 12329, 13049, 13499, 13859, 14159, 14489, 14519, 14639, 14669, ...}

Primes for which the weight as defined in A117078 is 23. (Cf. A119504(n), n ≥ 1)

{631, 773, 2467, 2833, 3121, 3203, 3347, 3617, 4219, 4733, 4909, 4951, 5273, 6619, 7027, 7129, 7529, 8263, 8783, 9049, 9413, 9643, 9649, 10891, 11483, 11719, 12541, 13093, 13183, 13841, ...}

Primes for which the weight as defined in A117078 is 53 and the gap or jump as defined in A001223 is 52. (Cf. A118924(n), n ≥ 1)

{19609, 547171, 3099757, 3282289, 3401221, 4286851, 4648099, 5544859, 5622769, 5731207, 5868901, 6387559, 6581857, 6949147, 6985081, 7382899, 7412791, 7675141, 7697401, 8203021, ...}

## Sequences related to primes classified by level

Primes classified by level. (Cf. A162174(n), n ≥ 1)

{5, 13, 19, 23, 31, 37, 43, 47, 53, 61, 73, 97, 113, 127, 131, 139, 151, 157, 163, 173, 181, 199, 211, 223, 233, 257, 263, 271, 293, 307, 313, 317, 337, 353, 373, 389, 397, 401, 421, 457, 479, 509, 523, ...}

Least prime of level 2n-1 (Cf. A117563 for level). (Cf. A118122(n), n ≥ 1)

{5, 11, 17, 509, 29, 83, 41, 79, 887, 59, 109, 71, 331, 193, 383, 190717, 101, 107, 787, 277, 1129, 911, 137, 1181, 149, 463, 1013, 839, 1087, 179, 433, 191, 197, 4093, 349, 503, 2423, 227, 701, 239, ...}

Primes for which the level is equal to 1 in A117563. (Cf. A125830(n), n ≥ 1)

{5, 13, 23, 31, 47, 53, 73, 157, 173, 211, 233, 257, 263, 353, 373, 563, 593, 607, 619, 647, 653, 733, 947, 977, 1069, 1097, 1103, 1123, 1187, 1223, 1283, 1367, 1433, 1453, 1459, 1493, 1499, 1511, ...}

Primes for which the level is equal to 3 in A117563. (Cf. A117873(n), n ≥ 1)

{11, 19, 37, 43, 97, 113, 127, 139, 163, 223, 307, 313, 317, 337, 389, 397, 401, 421, 457, 479, 547, 673, 691, 709, 757, 761, 853, 863, 883, 929, 937, 953, 1021, 1051, 1109, 1297, 1303, 1327, 1399, ...}

Primes for which the level is equal to 5 in A117563. (Cf. A117874(n), n ≥ 1)

{17, 61, 131, 151, 271, 523, 541, 571, 751, 797, 971, 991, 997, 1291, 1321, 1361, 1741, 1901, 1913, 2011, 2179, 2297, 2341, 2441, 2447, 2551, 2791, 2851, 3301, 3511, 3761, 3803, 4051, ...}

Primes for which the level is equal to 9 in A117563. (Cf. A118481(n), n ≥ 1)

{29, 67, 89, 181, 293, 811, 919, 1153, 1801, 2017, 2053, 2113, 2647, 3373, 3469, 3583, 4057, 5153, ...}

Primes for which the level is equal to 79 in A117563. (Cf. A118574(n), n ≥ 1)

{239, 719, 1033, 1193, 2143, 2777, 3889, 5953, 15917879, 16427897, 16754483, 24597451, 24612613, 27756503, 28261307, 28863287, 30493373, 30953633, 33444023, 34346203, 41488301, ...}

Primes p=prime(i) of level (1;1), i.e. such that A118534(i)=prime(i-1): balanced primes. (Cf. A006562(n), n ≥ 1)

{5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, ...}

Primes p=prime(i) of level (1;2), i.e. such that A118534(i)=prime(i-2). (Cf. A117876(n), n ≥ 1)

{23, 47, 73, 233, 353, 647, 1097, 1283, 1433, 1453, 1493, 1613, 1709, 1889, 2099, 2161, 2383, 2621, 2693, 2713, 3049, 3533, 3559, 3923, 4007, 4133, 4643, 4793, 4937, 5443, 5743, 6101, 7213, 7309, 7351, 7561, ...}

Primes p=prime(i) of level (1;3), i.e. such that A118534(i)=prime(i-3). (Cf. A118467(n), n ≥ 1)

{619, 1069, 1459, 1499, 1759, 1789, 2861, 3331, 3931, 4177, 4801, 4831, 5419, 6229, 6397, 8431, 8893, 9067, 9631, 11003, 11131, 11789, 12619, 14251, 15331, 15889, 16661, 17683, 17939, 18269, ...}

Primes p=prime(i) of level (1;5), i.e., such that A118534(i)=prime(i-5). (Cf. A118464(n), n ≥ 1)

{13933, 23633, 28229, 49223, 71363, 79633, 81239, 90547, 96857, 97613, 108827, 115363, 117443, 126781, 130657, 133733, 153533, 157679, 176819, 186799, 197389, 206651, 221327, 222199, ...}

Primes p=prime(i) of level (1;9), i.e., such that A118534(i)=prime(i-9). (Cf. A119404(n), n ≥ 1)

{678659, 855739, 1403981, 2366543, 2744783, 2830657, 3027539, 3317033, 4525909, 4676851, 5341463, 5819563, 7087123, 7181897, 8815663, 9324257, 9878929, 9976937, 10403251, 10440641, ...}

Primes p=prime(i) of level (1;10), i.e., such that A118534(i)=prime(i-10). (Cf. A119403(n), n ≥ 1)

{745757, 1103639, 1583369, 1895359, 2124049, 3327419, 4234537, 4437779, 5071973, 6287647, 7702573, 8470927, 8675923, 9493151, 9750079, 10868203, 11213843, 14244173, 14796253, ...}

Primes p=prime(i) of level (1;11), i.e., such that A118534(i)=prime(i-11). (Cf. A119402(n), n ≥ 1)

{576791, 3361517, 9433859, 10460719, 11630503, 11707537, 12080027, 19743677, 28716287, 33384517, 34961923, 36627659, 37776967, 38087983, 40794049, 45650359, 49152757, ...}

Primes p=prime(i) of level (1;12), i.e., such that A118534(i)=prime(i-12). (Cf. A125565(n), n ≥ 1)

{15014557, 27001043, 29602093, 50234633, 87028433, 91814759, 94529221, 103336843, 112840309, 113774329, 113961299, 114887657, 115528969, 118974901, 129235273, 144352123, 146127721, ...}

Primes p=prime(i) of level (1;13), i.e., such that A118534(i)=prime(i-13). (Cf. A125572(n), n ≥ 1)

{35630467, 118877047, 123823081, 140061577, 155032793, 175204303, 184606997, 188871349, 189489733, 232093339, 244004749, 278518081, 309055367, 310542257, 313596551, 315659909, ...}

Primes p=prime(i) of level (1;14), i.e., such that A118534(i)=prime(i-14). (Cf. A125574(n), n ≥ 1)

{31515413, 69730637, 132102911, 132375259, 215483129, 284491367, 325689253, 388190689, 548369603, 620829113, 633418787, 638213603, 670216277, 793852487, 797759539, 960200149, ...}

Primes p=prime(i) of level (1;15), i.e., such that A118534(i)=prime(i-15). (Cf. A125576(n), n ≥ 1)

{264426203, 295902073, 361949821, 704544167, ...}

Primes p=prime(i) of level (1;16), i.e., such that A118534(i)=prime(i-16). (Cf. A125623(n), n ≥ 1)

{356604959, 613768081, 709208323, 950803363, 979872743, ...}

# Sequences related to odd numbers

The odd numbers, $\scriptstyle a(n) \,=\, 2n+1,\ n \,\ge\, 0,\,$ (Cf. A005408) are

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, ...}

a(n) = a(1) = 1; for n>1, smallest divisor > 1 of 2n-1. (Cf. A090368(n), n ≥ 1) (referred to as the weight of odd numbers)

{1, 3, 5, 7, 3, 11, 13, 3, 17, 19, 3, 23, 5, 3, 29, 31, 3, 5, 37, 3, 41, 43, 3, 47, 7, 3, 53, 5, 3, 59, 61, 3, 5, 67, 3, 71, 73, 3, 7, 79, 3, 83, 5, 3, 89, 7, 3, 5, 97, 3, 101, 103, 3, 107, 109, 3, 113, 5, 3, 7, 11, 3, 5, ...}

a(n) = A005408(n-1) / A090368(n-1) for n > 2 and a(n) = 0 for n <= 2. (Cf. A184726(n), n ≥ 1) (referred to as the level of odd numbers)

{0, 0, 1, 1, 1, 3, 1, 1, 5, 1, 1, 7, 1, 5, 9, 1, 1, 11, 7, 1, 13, 1, 1, 15, 1, 7, 17, 1, 11, 19, 1, 1, 21, 13, 1, 23, 1, 1, 25, 11, 1, 27, 1, 17, 29, 1, 13, 31, 19, 1, 33, 1, 1, 35, 1, 1, 37, 1, 23, 39, 17, 11, 41, 25, ...}

# Sequences related to even numbers

The even numbers, $\scriptstyle a(n) \,=\, 2n,\ n \,\ge\, 0,\,$ (Cf. A005843) are

{0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, ...}

a(n) = Smallest divisor of 2n that is > 2, or 0 if no such divisor exists. (Cf. A090369(n), n ≥ 1) (referred to as the weight of even numbers)

{0, 4, 3, 4, 5, 3, 7, 4, 3, 4, 11, 3, 13, 4, 3, 4, 17, 3, 19, 4, 3, 4, 23, 3, 5, 4, 3, 4, 29, 3, 31, 4, 3, 4, 5, 3, 37, 4, 3, 4, 41, 3, 43, 4, 3, 4, 47, 3, 7, 4, 3, 4, 53, 3, 5, 4, 3, 4, 59, 3, 61, 4, 3, 4, 5, 3, 67, 4, ...}

a(n) = a(n) = A005843(n-1) / A090369(n-1) for n > 2 and a(n) = 0 for n <= 2. (Cf. A184727(n), n ≥ 1) (referred to as the level of even numbers)

{0, 0, 1, 2, 2, 2, 4, 2, 4, 6, 5, 2, 8, 2, 7, 10, 8, 2, 12, 2, 10, 14, 11, 2, 16, 10, 13, 18, 14, 2, 20, 2, 16, 22, 17, 14, 24, 2, 19, 26, 20, 2, 28, 2, 22, 30, 23, 2, 32, 14, 25, 34, 26, 2, 36, 22, 28, 38, 29, 2, 40, ...}

# Sequences related to composite numbers

The composite numbers: numbers n of the form x*y for x > 1 and y > 1. (Cf. A002808(n), n ≥ 1)

{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, ...}

Differences between composite numbers. (Cf. A073783(n), n ≥ 1)

{2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, ...}

a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists. (Cf. A130882(n), n ≥ 1) (referred to as the weight of composite numbers)

{0, 4, 7, 2, 4, 5, 13, 2, 7, 4, 19, 2, 4, 23, 2, 5, 2, 13, 4, 31, 2, 3, 2, 17, 37, 2, 19, 4, 43, 2, 4, 47, 2, 7, 2, 5, 53, 2, 5, 2, 4, 29, 61, 2, 3, 2, 4, 67, 2, 4, 5, 73, 2, 3, 2, 4, 79, 2, 4, 83, 2, 5, 2, 43, 89, 2, 7, 2, 3, 2, 47, 97, ...}

a(n) = A179620(n) / A130882(n) unless A130882(n) = 0 in which case a(n) = 0. (Cf. A179621(n), n ≥ 1) (referred to as the level of composite numbers)

{0, 1, 1, 4, 2, 2, 1, 7, 2, 4, 1, 10, 5, 1, 12, 5, 13, 2, 7, 1, 16, 11, 17, 2, 1, 19, 2, 10, 1, 22, 11, 1, 24, 7, 25, 10, 1, 27, 11, 28, 14, 2, 1, 31, 21, 32, 16, 1, 34, 17, 14, 1, 37, 25, 38, 19, 1, 40, 20, 1, 42, 17, 43, ...}

a(n) = largest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists. (Cf. A179620(n), n ≥ 1) (referred to as l(n) of composite numbers)

{0, 4, 7, 8, 8, 10, 13, 14, 14, 16, 19, 20, 20, 23, 24, 25, 26, 26, 28, 31, 32, 33, 34, 34, 37, 38, 38, 40, 43, 44, 44, 47, 48, 49, 50, 50, 53, 54, 55, 56, 56, 58, 61, 62, 63, 64, 64, 67, 68, 68, 70, 73, 74, 75, 76, 76, 79, 80, 80 ...}

## Sequences related to biprimes

Semiprimes (or biprimes): products of two primes. (Cf. A001358(n), n ≥ 1)

{4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, ...}

a(n) = Differences between products of 2 primes. (Cf. A065516(n), n ≥ 1)

{2, 3, 1, 4, 1, 6, 1, 3, 1, 7, 1, 1, 3, 1, 7, 3, 2, 4, 2, 1, 4, 3, 4, 5, 3, 5, 3, 1, 1, 4, 2, 1, 1, 11, 5, 4, 3, 1, 2, 1, 1, 6, 4, 1, 7, 1, 1, 2, 1, 9, 3, 1, 2, 5, 3, 8, 1, 5, 2, 2, 7, 7, 1, 1, 2, 1, 3, 4, 1, 1, 2, ...}

a(n) = smallest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists. (Cf. A130533(n), n ≥ 1) (referred to as the weight of biprimes, or weight of semiprimes, , or weight of 2-almost primes)

{0, 0, 2, 6, 13, 9, 2, 19, 2, 19, 2, 3, 4, 37, 8, 43, 47, 47, 53, 2, 6, 59, 61, 8, 71, 6, 79, 2, 5, 83, 89, 2, 3, 12, 101, 107, 4, 3, 3, 2, 11, 9, 5, 2, 127, 2, 3, 3, 2, 137, 4, 157, 157, 6, 163, 23, 2, 173, 181, 3, ...}

a(n) = largest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists. (Cf. A184728(n), n ≥ 1) (referred to as the l(n) of biprimes, or weight of semiprimes, , or weight of 2-almost primes)

{0, 0, 8, 6, 13, 9, 20, 19, 24, 19, 32, 33, 32, 37, 32, 43, 47, 47, 53, 56, 54, 59, 61, 64, 71, 72, 79, 84, 85, 83, 89, 92, 93, 84, 101, 107, 112, 117, 117, 120, 121, 117, 125, 132, 127, 140, 141, 141, 144, 137, ...}

a(n) = A184728(n) / A130533(n) unless A130533(n) = 0 in which case a(n) = 0. (Cf. A184729(n), n ≥ 1) (referred to as the level of biprimes, or level of semiprimes, , or level of 2-almost primes)

{0, 0, 4, 1, 1, 1, 10, 1, 12, 1, 16, 11, 8, 1, 4, 1, 1, 1, 1, 28, 9, 1, 1, 8, 1, 12, 1, 42, 17, 1, 1, 46, 31, 7, 1, 1, 28, 39, 39, 60, 11, 13, 25, 66, 1, 70, 47, 47, 72, 1, 38, 1, 1, 26, 1, 7, 88, 1, 1, 61, 20, 17, ...}

## Sequences related to triprimes

{8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, ...}

a(n) = smallest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists. (Cf. A130650(n), n ≥ 1) (referred to as the weight of triprimes, or weight of 3-almost primes)

{0, 0, 4, 13, 2, 13, 18, 4, 43, 8, 3, 41, 4, 4, 3, 13, 2, 37, 16, 43, 97, 4, 9, 10, 53, 4, 5, 10, 3, 6, 61, 43, 2, 11, 2, 12, 163, 8, 13, 2, 5, 173, 8, 89, 4, 3, 37, 61, 101, 101, 107, 229, 113, ...}

a(n) = largest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists. (Cf. A184752(n), n ≥ 1) (referred to as the l(n) of triprimes, or l(n) of 3-almost primes)

{0, 0, 16, 13, 26, 26, 18, 40, 43, 40, 48, 41, 60, 64, 66, 65, 74, 74, 64, 86, 97, 96, 99, 100, 106, 112, 115, 110, 123, 120, 122, 129, 146, 143, 152, 144, 163, 160, 169, 170, 170, 173, 168, 178, 184, 186, 185, ...}

a(n) = A184752(n) / A130650(n) unless A130650(n) = 0 in which case a(n) = 0. (Cf. A184753(n), n ≥ 1) (referred to as the level of triprimes, or level of 3-almost primes)

{0, 0, 4, 1, 13, 2, 1, 10, 1, 5, 16, 1, 15, 16, 22, 5, 37, 2, 4, 2, 1, 24, 11, 10, 2, 28, 23, 11, 41, 20, 2, 3, 73, 13, 76, 12, 1, 20, 13, 85, 34, 1, 21, 2, 46, 62, 5, 3, 2, 2, 2, 1, 2, 78, 39, 80, 81, 122, 3, ...}

# Sequences related to lucky numbers

Lucky numbers. (Cf. A000959(n), n ≥ 1)

{1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, ...}

First differences of lucky numbers. (Cf. A031883(n), n ≥ 1)

{2, 4, 2, 4, 2, 6, 4, 6, 2, 4, 6, 6, 2, 12, 4, 2, 4, 2, 4, 8, 6, 6, 6, 6, 4, 12, 2, 4, 2, 6, 10, 8, 4, 6, 2, 18, 4, 2, 6, 4, 6, 8, 4, 8, 4, 2, 4, 18, 2, 6, 6, 10, 2, 4, 8, 6, 4, 12, 2, 6, 4, 8, 10, 8, 4, 6, ...}

a(n) = smallest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists. (Cf. A130889(n), n ≥ 1) (referred to as the weight of lucky numbers)

{0, 0, 5, 5, 11, 9, 17, 19, 29, 29, 31, 37, 47, 13, 59, 5, 5, 71, 71, 71, 9, 29, 31, 9, 107, 103, 5, 5, 131, 43, 131, 11, 5, 157, 167, 51, 5, 191, 7, 197, 199, 29, 5, 43, 227, 233, 233, 223, 257, 15, 9, 263, ...}

a(n) = A184827(n) / A130889(n) unless A130889(n) = 0 in which case a(n) = 0. (Cf. A184828(n), n ≥ 1) (referred to as the level of lucky numbers)

{0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 13, 13, 1, 1, 1, 9, 3, 3, 11, 1, 1, 25, 25, 1, 3, 1, 13, 31, 1, 1, 3, 37, 1, 27, 1, 1, 7, 43, 5, 1, 1, 1, 1, 1, 17, 29, 1, 1, 1, 1, 3, 23, 5, 1, 45, 19, 19, 7, 31, ...}

a(n) = largest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists. (Cf. A184827(n), n ≥ 1) (referred to as the l(n) of lucky numbers)

{0, 0, 5, 5, 11, 9, 17, 19, 29, 29, 31, 37, 47, 39, 59, 65, 65, 71, 71, 71, 81, 87, 93, 99, 107, 103, 125, 125, 131, 129, 131, 143, 155, 157, 167, 153, 185, 191, 189, 197, 199, 203, 215, 215, 227, 233, ...}

# Sequences related to prime powers numbers

Prime powers p^k (p prime, k >= 0). . (Cf. A000961(n), n ≥ 1)

{1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, ...}

First differences of sequence of consecutive prime powers. (Cf. A057820(n), n ≥ 1)

{1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, 4, 2, 6, 2, 2, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 6, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, ...}

a(n) = smallest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists. (Cf. A184829(n), n ≥ 1) (referred to as the weight of prime powers)

{0, 0, 2, 3, 3, 2, 7, 7, 3, 5, 3, 3, 5, 3, 23, 5, 3, 2, 9, 11, 3, 13, 3, 5, 47, 3, 29, 61, 7, 3, 67, 7, 79, 7, 9, 31, 3, 9, 3, 5, 15, 9, 3, 2, 5, 25, 3, 43, 3, 29, 151, 53, 3, 5, 167, 3, 19, 3, 7, 3, 17, 199, 73, ...}

a(n) = A184829(n) / A130889(n) unless A130889(n) = 0 in which case a(n) = 0. (Cf. A184831(n), n ≥ 1) (referred to as the level of prime powers)

{0, 0, 1, 1, 1, 3, 1, 1, 3, 2, 5, 5, 3, 7, 1, 5, 9, 15, 3, 3, 13, 3, 15, 9, 1, 19, 2, 1, 9, 23, 1, 11, 1, 11, 9, 3, 33, 11, 35, 21, 7, 13, 41, 63, 25, 5, 45, 3, 49, 5, 1, 3, 55, 33, 1, 59, 9, 63, 27, 65, ...}

a(n) = largest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists. (Cf. A184830(n), n ≥ 1) (referred to as the l(n) of prime powers)

{0, 0, 2, 3, 3, 6, 7, 7, 9, 10, 15, 15, 15, 21, 23, 25, 27, 30, 27, 33, 39, 39, 45, 45, 47, 57, 58, 61, 63, 69, 67, 77, 79, 77, 81, 93, 99, 99, 105, 105, 105, 117, 123, 126, 125, 125, 135, 129, 147, 145, 151, ...}

# Sequences related to squarefree numbers

Squarefree numbers: numbers that are not divisible by a square greater than 1. (Cf. A005117(n), n ≥ 1)

{1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, ...}

Gaps between squarefree numbers. (Cf. A076259(n), n ≥ 1)

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

a(n) = smallest k such that A005117(n+1) = A005117(n) + (A005117(n) mod k), or 0 if no such k exists. (Cf. A184832(n), n ≥ 1) (referred to as the weight of squarefree numbers)

{0, 0, 0, 2, 5, 4, 3, 3, 2, 13, 13, 3, 17, 2, 3, 4, 23, 2, 29, 29, 2, 3, 3, 2, 37, 37, 2, 41, 4, 3, 43, 7, 3, 53, 2, 3, 3, 2, 59, 2, 5, 5, 2, 3, 3, 2, 71, 2, 7, 4, 3, 3, 2, 5, 5, 3, 89, 2, 3, 3, 31, 2, 101, 101, ...}

a(n) = A184833(n) / A184832(n) unless A184832(n) = 0 in which case a(n) = 0. (Cf. A184834(n), n ≥ 1) (referred to as the level of squarefree numbers)

{0, 0, 0, 2, 1, 1, 3, 3, 6, 1, 1, 5, 1, 10, 7, 5, 1, 14, 1, 1, 16, 11, 11, 18, 1, 1, 20, 1, 10, 15, 1, 7, 17, 1, 28, 19, 19, 30, 1, 32, 13, 13, 34, 23, 23, 36, 1, 38, 11, 19, 27, 27, 42, 17, 17, 29, 1, 46, 31, ...}

a(n) = largest k such that A005117(n+1) = A005117(n) + (A005117(n) mod k), or 0 if no such k exists. (Cf. A184833(n), n ≥ 1) (referred to as the l(n) of squarefree numbers)

{0, 0, 0, 4, 5, 4, 9, 9, 12, 13, 13, 15, 17, 20, 21, 20, 23, 28, 29, 29, 32, 33, 33, 36, 37, 37, 40, 41, 40, 45, 43, 49, 51, 53, 56, 57, 57, 60, 59, 64, 65, 65, 68, 69, 69, 72, 71, 76, 77, 76, 81, 81, 84, ...}

# Sequences related to figurate numbers

## Sequences related to triangular numbers

Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. (Cf. A000217(n), n ≥ 0)

{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, ...}

$\scriptstyle d(n) \,=\, P^{(2)}_{3}(n+1) - P^{(2)}_{3}(n) \,=\, n+1 \,$ (Cf. A000027(n+1), n ≥ 0)

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, ...}

a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists. (Cf. A130703(n), n ≥ 1) (referred to as the weight of triangular numbers)

{0, 0, 0, 0, 9, 14, 10, 27, 35, 22, 18, 65, 77, 18, 26, 119, 27, 38, 34, 27, 209, 46, 28, 55, 299, 36, 35, 377, 45, 62, 58, 45, 527, 40, 54, 629, 95, 54, 74, 779, 63, 86, 82, 63, 989, 94, 54, 161, 235, 68, 91, ...}

a(n) = A184218(n) / A130703(n) unless A130703(n) = 0 in which case a(n) = 0. (Cf. A184219(n), n ≥ 1) (referred to as the level of triangular numbers)

{0, 0, 0, 0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 5, 4, 1, 5, 4, 5, 7, 1, 5, 9, 5, 1, 9, 10, 1, 9, 7, 8, 11, 1, 14, 11, 1, 7, 13, 10, 1, 13, 10, 11, 15, 1, 11, 20, 7, 5, 18, 14, 5, 17, 22, 14, 19, 11, 14, 19, 1, 1, 27, 16, 13, 27, 16 ...}

a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists. (Cf. A184218(n), n ≥ 1) (referred to as l(n) of triangular numbers)

a(n) = (n+1)*(n-2)/2 = A000096(n-2) for n ≥ 5 and a(n) = 0 for n ≤ 4.

{0, 0, 0, 0, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080 ...}

## Sequences related to square numbers

Square numbers: a(n) = n^2. (Cf. A000290(n), n ≥ 0)

{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, ...}

$\scriptstyle d(n) \,=\, P^{(2)}_{4}(n+1) - P^{(2)}_{4}(n) \,=\, 2n+1 \,$ (Cf. A005408(n), n ≥ 0)

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, ...}

a(n) = smallest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists. (Cf. A133150(n), n ≥ 1) (referred to as the weight of squares)

{0, 0, 0, 0, 14, 23, 17, 47, 31, 79, 49, 119, 71, 167, 97, 223, 127, 41, 46, 359, 199, 439, 241, 527, 82, 89, 337, 727, 391, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 217, 94, 1679, 881, 1847, ...}

a(n) = A184220(n) / A133150(n) unless A133150(n) = 0 in which case a(n) = 0. (Cf. A184221(n), n ≥ 1) (referred to as the level of square numbers)

{0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 7, 7, 1, 2, 1, 2, 1, 7, 7, 2, 1, 2, 1, 2, 7, 14, 1, 2, 1, 2, 1, 14, 7, 17, 1, 2, 1, 2, 17, 14, 1, 2, 1, 2, 23, 14, 7, 2, 1, 2, 17, 2, 7, 14, 1, 17, 1, 23, 1, 14, 7, 2, 1, 31 ...}

a(n) = largest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists. (Cf. A184220(n), n ≥ 1) (referred to as l(n) of square numbers)

a(n) = (n-1)^2-2 = A008865(n-1) for n ≥ 5 and a(n) = 0 for n ≤ 4.

{0, 0, 0, 0, 14, 23, 34, 47, 62, 79, 98, 119, 142, 167, 194, 223, 254, 287, 322, 359, 398, 439, 482, 527, 574, 623, 674, 727, 782, 839, 898, 959, 1022, 1087, 1154, 1223, 1294, 1367, 1442, 1519, 1598, 1679 ...}

## Sequences related to pentagonal numbers

Pentagonal numbers: n(3n-1)/2. (Cf. A000326(n), n ≥ 0)

{0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, ...}

$\scriptstyle d(n) \,=\, P^{(2)}_{5}(n+1) - P^{(2)}_{5}(n) \,=\, 3n+1 \,$ (Cf. A016777 (n+1), n ≥ 0)

{1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, ...}

a(n) = smallest k such that A000326(n+1) = A000326(n) + (A000326(n) mod k), or 0 if no such k exists. (Cf. A133151(n), n ≥ 1) (referred to as the weight of pentagonal numbers)

{0, 0, 0, 0, 19, 32, 24, 67, 89, 38, 71, 173, 69, 61, 71, 109, 373, 211, 79, 529, 587, 72, 89, 779, 283, 461, 499, 359, 1159, 311, 111, 1423, 1517, 269, 857, 1817, 641, 127, 134, 251, 2377, 1249, 138, ...}

a(n) = A184750(n) / A133151(n) unless A133151(n) = 0 in which case a(n) = 0. (Cf. A184751(n), n ≥ 1) (referred to as the level of pentagonal numbers)

{0, 0, 0, 0, 1, 1, 2, 1, 1, 3, 2, 1, 3, 4, 4, 3, 1, 2, 6, 1, 1, 9, 8, 1, 3, 2, 2, 3, 1, 4, 12, 1, 1, 6, 2, 1, 3, 16, 16, 9, 1, 2, 19, 1, 1, 12, 4, 19, 9, 2, 2, 3, 1, 8, 24, 1, 1, 18, 23, 1, 3, 19, 4, ...}

a(n) = largest k such that A000326(n+1) = A000326(n) + (A000326(n) mod k), or 0 if no such k exists. (Cf. A184750(n), n ≥ 1) (referred to as l(n) of pentagonal numbers)

a(n) = 3(n^2-7n-2)/2 for n ≥ 5 and a(n) = 0 for n ≤ 4.

{0, 0, 0, 0, 19, 32, 48, 67, 89, 114, 142, 173, 207, 244, 284, 327, 373, 422, 474, 529, 587, 648, 712, 779, 849, 922, 998, 1077, 1159, 1244, 1332, 1423, 1517, 1614, 1714, 1817, 1923, 2032, 2144, 2259, ...}