A223703
Conjectured irregular triangle (with some rows blank) of numbers k such that prime(n) is the largest prime factor of k^3 - 1.
Original entry on oeis.org
2, 4, 3, 9, 16, 22, 18, 7, 11, 30, 5, 25, 67, 191, 10, 26, 100, 121, 211, 581, 676, 6, 36, 49, 79, 87, 165, 6205, 178, 13, 47, 501, 562, 29, 37, 68, 135, 163, 565, 900, 1369, 1712, 3446, 4624, 8, 64, 74, 81, 137, 373, 439, 1451, 1816, 2629, 7527, 39209
Offset: 1
Irregular triangle:
2: {},
3: {},
5: {},
7: {2, 4},
11: {},
13: {3, 9, 16, 22},
17: {18},
19: {7, 11},
23: {},
29: {30},
31: {5, 25, 67, 191},
37: {10, 26, 100, 121, 211, 581, 676},
41: {},
43: {6, 36, 49, 79, 87, 165},
47: {6205},
53: {},
59: {178},
61: {13, 47, 501, 562},
67: {29, 37, 68, 135, 163, 565, 900, 1369, 1712, 3446, 4624},
71: {},
73: {8, 64, 74, 81, 137, 373, 439, 1451, 1816, 2629, 7527, 39209}
Cf.
A175607 (largest number k such that the greatest prime factor of k^2-1 is prime(n)).
-
t = Table[FactorInteger[n^3 - 1][[-1,1]], {n, 2, 10^6}]; Table[1 + Flatten[Position[t, Prime[n]]], {n, 25}]
A223704
Conjectured irregular triangle (with some rows blank) of numbers k such that prime(n) is the largest prime factor of k^3 + 1.
Original entry on oeis.org
1, 2, 3, 5, 19, 4, 10, 17, 23, 8, 12, 31, 69, 6, 26, 68, 11, 27, 101, 122, 7, 37, 50, 80, 179, 582, 14, 48, 75, 563, 719, 2820, 4135, 30, 38, 164, 231, 440, 566, 901, 11093, 112925, 267167, 212, 9, 65, 374, 20303, 24, 56, 103, 293, 530, 656, 767, 868, 82, 2157
Offset: 1
Irregular triangle:
2: {1},
3: {2},
5: {},
7: {3, 5, 19},
11: {},
13: {4, 10, 17, 23},
17: {},
19: {8, 12, 31, 69},
23: {},
29: {},
31: {6, 26, 68},
37: {11, 27, 101},
41: {122},
43: {7, 37, 50, 80, 179},
47: {},
53: {582},
59: {},
61: {14, 48, 75, 563, 719, 2820, 4135},
67: {30, 38, 164, 231, 440, 566, 901, 11093, 112925, 267167},
71: {212},
73: {9, 65, 374, 20303},
79: {24, 56, 103, 293, 530, 656, 767, 868},
83: {82, 2157}.
Cf.
A175607 (largest number k such that the greatest prime factor of k^2-1 is prime(n)).
-
t = Table[FactorInteger[n^3 + 1][[-1,1]], {n, 10^6}]; Table[Flatten[Position[t, Prime[n]]], {n, 25}]
A223705
Least number k such that prime(n) is the largest divisor of k^2 + 1, or 0 if there is no such k.
Original entry on oeis.org
1, 0, 2, 0, 0, 5, 4, 0, 0, 12, 0, 6, 9, 0, 0, 23, 0, 11, 0, 0, 27, 0, 0, 34, 22, 10, 0, 0, 33, 15, 0, 0, 37, 0, 44, 0, 28, 0, 0, 80, 0, 19, 0, 81, 14, 0, 0, 0, 0, 107, 89, 0, 64, 0, 16, 0, 82, 0, 60, 53, 0, 138, 0, 0, 25, 114, 0, 148, 0, 136, 42, 0, 0, 104, 0, 0
Offset: 1
-
nn = 100; t = Table[0, {nn}]; Do[If[Mod[Prime[n], 4] == 3, t[[n]] = -1], {n, nn}]; n = 0; While[Times @@ t == 0, n++; s = FactorInteger[n^2 + 1][[-1, 1]]; p = PrimePi[s]; If[p <= nn && t[[p]] == 0, t[[p]] = n]]; Do[If[Mod[Prime[n], 4] == 3, t[[n]] = 0], {n, nn}]; t
A223706
Conjectured least number k such that prime(n) is the largest divisor of k^3 - 1, or 0 if there is no such k.
Original entry on oeis.org
0, 0, 0, 2, 0, 3, 18, 7, 0, 30, 5, 10, 0, 6, 6205, 0, 178, 13, 29, 0, 8, 23, 0, 0, 35, 102, 46, 429, 45, 0, 19, 263, 0, 42, 7600, 32, 12, 58, 1083163, 520, 0, 48, 880129, 84, 5320, 92, 14, 39, 13848, 94, 4195, 718, 15, 21085, 772, 502068, 5381, 28, 116, 5579256
Offset: 1
-
nn = 60; t = Table[0, {nn}]; ps = Prime[Range[nn]]; Do[num = n^3 - 1; j = 0; lastP = 0; While[num > 0 && j < nn, j++; p = ps[[j]]; While[Mod[num, p] == 0, lastP = j; num = num/p]];If[num == 1 && t[[lastP]] == 0, t[[lastP]] = n; Print[{lastP, n}]], {n, 10^7}]; t
Showing 1-4 of 4 results.
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