| AA04418 |
|
|
| Data | 3, 27, 63, 75, 99, 135, 147, 171, 195, 207, 231, 243, 255, 279, 315, 351, 363, 375, 387, 399, 423, 435, 459, 483, 495, 507, 531, 555, 567, 603, 615, 627, 639, 651, 663, 675, 711, 735, 747, 759, 783, 795, 819, 855, 867, 891, 903, 915, 927, 963, 975, 987, 999, 1023, 1035, 1071, 1083, 1095, 1107, 1131, 1143, 1155, 1179, 1215, 1239, 1251, 1275, 1287, 1311, 1323, 1335, 1359, 1395, 1407, 1419, 1431, 1443, 1455, 1467, 1479, 1491, 1503, 1515, 1539, 1551, 1575, 1587, 1599, 1611, 1635, 1647, 1659, 1683, 1695, 1719, 1743, 1755, 1767, 1791, 1815, 1827, 1863, 1875, 1887, 1899, 1911, 1935, 1947, 1971, 1995, 2007, 2043, 2055, 2067, 2079, 2091, 2115, 2139, 2151, 2163, 2175, 2187, 2211, 2223, 2235, 2247, 2259, 2295, 2331, 2343, 2355, 2367, 2379, 2403, 2415, 2439, 2451, 2475, 2499, 2511, 2523, 2535, 2547, 2583, 2595, 2607, 2619, 2655, 2667, 2679, 2691, 2703, 2715, 2727, 2739, 2751, 2763, 2775, 2799, 2835, 2847, 2871, 2883, 2895, 2907, 2919, 2943, 2955, 2967, 2979, 3003, 3015, 3051, 3075, 3087, 3111, 3123, 3135, 3159, 3171, 3195, 3219, 3231, 3243, 3255, 3267, 3303, 3315, 3339, 3363, 3375, 3399, 3411, 3423, 3435, 3447, 3471, 3483, 3495, 3507, 3519, 3531, 3555, 3567, 3591 | |
| Offset | 1,1 | |
| Comments | ||
| Links | ||
| Formula | a(n) =3*A091113(n) | |
| Example | ||
| Mathematica | 3*Select[4*Range[0,100]+1,!PrimeQ[#]&] | |
| Prog. (Magma) | [(((4*n+1)-n)*4)-1: n in [0..300]| not IsPrime(4*n+1) ]; | |
| Crossrefs | Oeis – A002144 , A118236 – A091113 | |
| Author | Pietro Maiorana Montes Mar 16 2020 | |
| Formula ad program by Vincenzo Librandi Mar 17 2020 |
AA04417 a(n) = (((4*n+1)-n)*4)-1. With prime(4*n+1) ];
| AA04417 |
|
|
| Data | 15, 39, 51, 87, 111, 123, 159, 183, 219, 267, 291, 303, 327, 339, 411, 447, 471, 519, 543, 579, 591, 687, 699, 723, 771, 807, 831, 843, 879, 939, 951, 1011, 1047, 1059, 1119, 1167, 1191, 1203, 1227, 1263, 1299, 1347, 1371, 1383, 1527, 1563, 1623, 1671, 1707, 1731, 1779, 1803, 1839, 1851, 1923, 1959, 1983, 2019, 2031, 2103, 2127, 2199, 2271, 2283, 2307, 2319, 2391, 2427, 2463, 2487, 2559, 2571, 2631, 2643, 2787, 2811, 2823, 2859, 2931, 2991, 3027, 3039, 3063, 3099, 3147, 3183, 3207, 3279, 3291, 3327, 3351, 3387, 3459, 3543, 3579, 3603 | |
| Offset | 1,1 | |
| Comments | ||
| Links | ||
| Formula | a(n) = 3*A002144(n) | |
| Example | ||
| Mathematica | 3 * Select[Range[5,617,4],PrimeQ] | |
| Prog. (Magma) | [(((4*n+1)-n)*4)-1: n in [0..300]| IsPrime(4*n+1) ]; | |
| Crossrefs | Oeis - A002144 , A118236 | |
| Author | Pietro Maiorana Montes Mar 16 2020 | |
| Formula and program by Vincenzo Librandi Mar 17 2020 |
AA04416 Semiprimes that are sum of 2, 3, and 4 consecutive semiprimes
| AA04416 |
|
|
| Data | 2045, 2705, 2855, 14614, 18838, 28437, 31299, 43603, 68807, 76841, 77386, 88041, 108415, 116822, 194605 | |
| Offset | 1,1 | |
| Comments | Also sums of 5 consecutive semiprimes: 2705, 88041. | |
| Example | 2045 = 1018 +1027 = 679 + 681 + 685 = 505 + 511 + 514 + 515. | |
| Mathematica | ||
| Prog. (Magma) | ||
| Crossrefs | Cf. Z360 in facebook.com/zak.seidov | |
| Author | Zak Seidov Mar 12 2020 | |
AA04415 Primes p, q such that p + 2* q = 625 = 25^2.
| AA04415 |
|
|
| Data | {3, 311}, {11, 307}, {59, 283}, {71, 277}, {83, 271}, {167, 229}, {179, 223}, {227, 199}, {239, 193}, {263, 181}, {311, 157}, {347, 139}, {419, 103}, {431, 97}, {467, 79}, {479, 73}, {491, 67}, {503, 61}, {563, 31}, {587, 19}, {599, 13}, {619, 3} |
|
| Offset | 1,1 | |
| Comments | 22 terms with 44 primes – fini full. | |
| Example | 3+2*311 = 625 = 25^2 11+2*307 = 625 = 25^2 |
|
| Mathematica | ||
| Prog. (Magma) | ||
| Crossrefs | Cf. Z362 in facebook.com/zak.seidov | |
| Author | Zak Seidov Mar 13 2020 | |
AA04414 Primes: a(n+1) – a(n) is the smallest multiple of the last digit of a(n), with a(1) = 11.
| AA04414 |
|
|
| Data | 11, 13, 19, 37, 79, 97, 139, 157, 199, 271, 277, 347, 389, 443, 449, 467, 509, 563, 569, 587, 601, 607, 677, 691, 701, 709, 727, 769, 787, 829, 883, 907, 977, 991, 997, 1039, 1093, 1117, 1187, 1201, 1213, 1231, 1237, 1279, 1297, 1367, 1381, 1399, 1453, 1459, 1531, 1543, 1549, 1567, 1609, 1627, 1669, 1723, 1741, 1747, 1789, 1861, 1867, 1951, 1973, 1979, 1997, 2011, 2017, 2087, 2129, 2237, 2251, 2267, 2281, 2287, 2357, 2371, 2377, 2447, 2503, 2521, 2531, 2539, 2557, 2683, 2689, 2707, 2749, 2767, 2837, 2851, 2857, 2927, 2969, 3023, 3041, 3049, 3067, 3109, 3163, 3169, 3187, 3229, 3301, 3307, 3391, 3407, 3449, 3467, 3593, 3617, 3631, 3637, 3833, 3851, 3853, 3877, 3919, 4027, 4111, 4127, 4211, 4217, 4231, 4241, 4243, 4261, 4271, 4273, 4297, 4339, 4357, 4441, 4447, 4517, 4643, 4649, 4703, 4721, 4723, 4729, 4783, 4789, 4861, 4871, 4877, 4919, 4937, 4951, 4957, 4999, 5107, 5233, 5281, 5297, 5381, 5387, 5443, 5449, 5503, 5521, 5527, 5569, 5623, 5641, 5647, 5689, 5743, 5749, 5821, 5827, 5869, 5923, 5953, 6007, 6091, 6101, 6113, 6131, 6133, 6151, 6163, 6199, 6217, 6287, 6301, 6311, 6317, 6359, 6449, 6521, 6529, 6547, 6659, 6803, 6827, 6841, 6857, 6871, 6883, 6907 | |
| Offset | 1,1 | |
| Comments | ||
| Formula | a(n) == 1 mod 10. | |
| Example | 13 – 11 = 2 = 2*1, 19 -13 = 6 = 2*3, 37 – 19 = 18 = 2*9, 79 – 37 = 42 = 6*7. Minimal difference is 2 when a(n) and a(n+1) are twin primes |
|
| Mathematica | s={11};p=11;Do[d=Mod[p,10];q=NextPrime[p];While[Mod[q-p,d]>0,q=NextPrime[q]]; AppendTo[s,p=q],{200}];s | |
| Prog. (Magma) | ||
| Crossrefs | Cf. Z361 in facebook.com/zak.seidov | |
| Author | Zak Seidov Mar 12 2020 | |
AA04413 Triangle in which n-th row lists all partitions of Prime(n), in graded reverse lexicographic ordering.
| AA04413 |
|
|
| Data | 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 5, 1, 1, 4, 3, 4, 2, 1, 4, 1, 1, 1, 3, 3, 1, 3, 2, 2, 3, 2, 1, 1, 3, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 | |
| Offset | 1,3 | |
| Example | {{2},{1,1}, {{3},{2,1},{1,1,1}}, {{5},{4,1},{3,2},{3,1,1},{2,2,1},{2,1,1,1},{1,1,1,1,1}}, |
|
| Mathematica | Table[IntegerPartitions[Prime[n]],{n,1,7}] | |
| Prog. (Magma) | &cat[&cat Partitions(NthPrime(n)):n in[1..4]]; | |
| Crossrefs | Oeis – A000045, A080577. | |
| Author | Vincenzo Librandi (vincenzo.librandi@tin.t) Jan 31 2020 | |
AA04412 Triangle in which n-th row lists all partitions of Fibonacci(n), in graded reverse lexicographic ordering.
| AA04412 |
|
|
| Data | 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 8, 7, 1, 6, 2, 6, 1, 1, 5, 3, 5, 2, 1, 5, 1, 1, 1, 4, 4, 4, 3, 1, 4, 2, 2, 4, 2, 1, 1, 4, 1, 1, 1, 1, 3, 3, 2, 3, 3, 1, 1, 3, 2, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 | |
| Offset | 1,3 | |
| Example | First five rows are: {{1}}, {{1}}, {{2},{1,1}}, {{3},{2,1},{1,1,1}}, {{5},{4,1},{3,2},{3,1,1},{2,2,1},{2,1,1,1},{1,1,1,1,1}} |
|
| Mathematica | Table[IntegerPartitions[Fibonacci[n]],{n,1,7}] | |
| Prog. (Magma) | &cat[&cat Partitions(Fibonacci(n)):n in[1..6]]; | |
| Crossrefs | Oeis – A000045, A080577. | |
| Author | Vincenzo Librandi (vincenzo.librandi@tin.t) Jan 31 2020 | |
AA04411 Primes with new semiprime gaps: a(1) = 2; afterwards a(n) is the smallest prime p > a(n-1) such that p – a(n-1) is a new semiprime.
| AA04411 |
|
|
| Data | 2, 11, 17, 31, 41, 67, 71, 109, 131, 193, 227, 313, 359, 433, 491, 613, 719, 853, 947, 1093, 1259, 1453, 1571, 1777, 1979, 2137, 2351, 2677, 2819, 3037, 3119, 3373, 3671, 4057, 4283, 4561, 4919, 5233, 5507, 5869, 6047, 6469, 6803, 7369, 7823, 8221, 8699, 9001, 9467, 9949, 10211, 10657, 11003, 11617, 12011, 12553, 13187, 13921, 14303, 14929, 15443, 15901, 16427, 16981, 17483, 18181, 18719, 19381, 20147, 21013, 21599, 22273, 23159, 23917, 24611, 25357, 25919, 26713, 27431, 28309, 29147, 29989, 30851, 31849, 32771, 33589, 34211, 35257, 35963, 36877, 37811, 38737, 39719, 40693, 41651, 42793, 43691, 44773, 45779, 46933, 47711, 48973, 49991, 51193, 52379, 53593, 54767, 56053, 57179, 58417, 59219, 60601, 61643, 62869, 64007, 65101, 66383, 67801, 69119, 70573, 71807, 73309, 74507, 75853, 77291, 79273, 80387, 81853, 83207, 84913, 86399, 87721, 89087, 90709, 92003, 93481, 95003, 96517, 97919, 99577, 101399, 103237, 104543, 106189, 107843, 109597, 111143, 112909, 114671, 116689, 118463, 120181, 122039, 123853, 125471, 127549, 129263, 131437, 133319, 134857, 136751, 138793, 140759, 142897, 144539, 146941, 148667, 150769, 152363, 154621, 156659, 158233, 159911, 161977, 163883, 165817, 167759, 169633, 171659, 174121, 176243, 178489, 181211, 183397, 185699, 188197, 190391, 192697, 195023, 197257, 199211, 201337, 203771, 206197, 208379, 210853, 213407, 216061, 218279, 220861, 223247, 225241, 227303 | |
| Offset | 1,1 | |
| Comments | Corresponding difference (all new semiprimes): 9, 6, 14, 10, 26, 4, 38, 22, 62, 34, 86, 46, 74, 58, 122, 106, 134, 94, 146, 166, 194, 118, 206, 202, 158, 214, 326, 142, 218, 82, 254, 298, 386, 226, 278, 358, 314, 274, 362, 178, 422, 334, 566, 454, 398, 478, 302, 466, 482, 262, 446, 346, 614, 394, 542, 634, 734, 382, 626, 514, 458, 526, 554, 502, 698, 538, 662, 766, 866, 586, 674, 886, 758, 694, 746, 562, 794, 718, 878, 838, 842, 862, 998, 922, 818, 622, 1046, 706, 914, 934, 926, 982, 974, 958, 1142, 898, 1082, 1006, 1154, 778, 1262, 1018, 1202, 1186, 1214, 1174, 1286, 1126, 1238, 802, 1382, 1042, 1226, 1138, 1094, 1282, 1418, 1318, 1454, 1234, 1502, 1198, 1346, 1438, 1982, 1114, 1466, 1354, 1706, 1486, 1322, 1366, 1622, 1294, 1478, 1522, 1514, 1402, 1658, 1822, 1838, 1306, 1646, 1654, 1754, 1546, 1766, 1762, 2018, 1774, 1718, 1858, 1814, 1618, 2078, 1714, 2174, 1882, 1538, 1894, 2042, 1966, 2138, 1642, 2402, 1726, 2102, 1594, 2258, 2038, 1574, 1678, 2066, 1906, 1934, 1942, 1874, 2026, 2462, 2122, 2246, 2722, 2186, 2302, 2498, 2194, 2306, 2326, 2234, 1954, 2126, 2434, 2426, 2182, 2474, 2554, 2654, 2218, 2582, 2386, 1994, 2062 |
|
| Example | ||
| Mathematica | sg={9,6}; s={2,11,17}; q=17; Do[p=q; q=NextPrime[p]; While[2!=PrimeOmega[d=q-p]||MemberQ[sg,d], q=NextPrime[q]]; AppendTo[s,q]; AppendTo[sg,d], {200}]; s | |
| Prog. (Magma) | ||
| Crossrefs | Cf. Z333 in facebook.com/zak.seidov | |
| Author | Zak Seidov Jan 09 2020 | |
AA04410 Primes with semiprime gaps
| AA04410 |
|
|
| Data | 2, 11, 17, 23, 29, 43, 47, 53, 59, 73, 79, 83, 89, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 179, 193, 197, 211, 233, 239, 277, 281, 307, 311, 317, 331, 337, 347, 353, 359, 373, 379, 383, 389, 463, 467, 541, 547, 557, 563, 569, 607, 613, 617, 631, 641, 647, 653, 659, 673, 677, 683, 709, 719, 733, 739, 743, 757, 761, 787, 797, 811, 821, 827, 853, 857, 863, 877, 881, 887, 1009, 1013, 1019, 1033, 1039, 1049, 1063, 1069, 1091, 1097, 1103, 1109, 1123, 1129, 1151, 1213, 1217, 1223, 1229, 1291, 1297, 1301, 1307, 1321, 1327, 1361, 1367, 1373, 1399, 1409, 1423, 1427, 1433, 1439, 1453, 1459, 1481, 1487, 1493, 1499, 1621, 1627, 1637, 1663, 1667, 1693, 1697, 1723, 1733, 1747, 1753, 1759, 1877, 1951, 1973, 1979, 1993, 1997, 2003, 2017, 2027, 2053, 2063, 2069, 2083, 2087, 2113, 2207, 2213, 2239, 2243, 2269, 2273, 2287, 2293, 2297, 2311, 2333, 2339, 2377, 2381, 2467, 2473, 2477, 2503, 2549, 2671, 2677, 2683, 2687, 2693, 2699, 2713, 2719, 2729, 2767, 2777, 2791, 2797, 2801, 2887, 2897, 2903, 2909, 2971, 3089, 3163, 3167, 3181, 3187, 3191, 3217, 3221, 3259, 3461, 3467, 3529, 3533, 3539, 3613 | |
| Offset | 1,1 | |
| Comments | Corresponding differences (all semiprimes): {9, 6, 6, 6, 14, 4, 6, 6, 14, 6, 4, 6, 14, 4, 6, 14, 4, 6, 14, 6, 6, 4, 6, 6, 14, 4, 14, 22, 6, 38, 4, 26, 4, 6, 14, 6, 10, 6, 6, 14, 6, 4, 6, 74, 4, 74, 6, 10, 6, 6, 38, 6, 4, 14, 10, 6, 6, 6, 14, 4, 6, 26, 10, 14, 6, 4, 14, 4, 26, 10, 14, 10, 6, 26, 4, 6, 14, 4, 6, 122, 4, 6, 14, 6, 10, 14, 6, 22, 6, 6, 6, 14, 6, 22, 62, 4, 6, 6, 62, 6, 4, 6, 14, 6, 34, 6, 6, 26, 10, 14, 4, 6, 6, 14, 6, 22, 6, 6, 6, 122, 6, 10, 26, 4, 26, 4, 26, 10, 14, 6, 6, 118, 74, 22, 6, 14, 4, 6, 14, 10, 26, 10, 6, 14, 4, 26, 94, 6, 26, 4, 26, 4, 14, 6, 4, 14, 22, 6, 38, 4, 86, 6, 4, 26, 46, 122, 6, 6, 4, 6, 6, 14, 6, 10, 38, 10, 14, 6, 4, 86, 10, 6, 6, 62, 118, 74, 4, 14, 6, 4, 26, 4, 38, 202, 6, 62, 4, 6, 74} |
|
| Example | ||
| Mathematica | ||
| Prog. (Magma) | ||
| Crossrefs | Cf. Z333 in facebook.com/zak.seidov | |
| Author | Zak Seidov Jan 09 2020 | |
AA04409 Indices of primes followed by gaps {0, 2, 4, 6, 8, 10, 12} mod 14.
| AA04409 |
|
|
| Data | 748141, 2007417, 3326778, 3613188, 3735906, 4081888, 4223498, 4270343, 4336388, 5054340, 5101217, 5245943, 5425674, 5476927, 5798981, 6739350, 7858778, 7939874, 7974478, 8640829, 9862393, 9925868 | |
| Offset | 1,1 | |
| Comments | Corresponding primes: 11351147, 32580593, 55793923, 60913367, 63118577, 69348373, 71905973, 72751949, 73948223, 87020567, 87879529, 90527867, 93826981, 94767319, 100692749, 118094953, 138998983, 140520349, 141172951, 153704387, 176812297, 178016009 |
|
| Example | prime(748141..748148) = {11351147, 11351161, 11351191, 11351237, 11351243, 11351251, 11351261, 11351287}; differences: {14, 30, 46, 6, 8, 10, 26} = {0, 2, 4, 6, 8, 10, 12} mod 14. |
|
| Mathematica | ||
| Prog. (Magma) | ||
| Crossrefs | Oeis – A320705 Indices of primes followed by a gap 14
Cf. Z332 in facebook.com/zak.seidov |
|
| Author | Zak Seidov Dec 30 2019 | |
SEQUENZE NUMERICHE LIBRANDI 