A064078
Zsigmondy numbers for a = 2, b = 1: Zs(n, 2, 1) is the greatest divisor of 2^n - 1 (A000225) that is coprime to 2^m - 1 for all positive integers m < n.
Original entry on oeis.org
1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681
Offset: 1
a(4) = 5 because 2^4 - 1 = 15 and its divisors being 1, 3, 5, 15, only 1 and 5 are coprime to 2^2 - 1 = 3 and 2^3 - 1 = 7, and 5 is the greater of these.
a(5) = 31 because 2^5 - 1 = 31 is prime.
a(6) = 1 because 2^6 - 1 = 63 and its divisors being 1, 3, 7, 9, 21, 63, only 1 is coprime to all of 3, 7, 15, 31.
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Table[Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]], {n, 40}] (* Alonso del Arte, Mar 14 2013 *)
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a(n) = my(m = polcyclo(n, 2)); m/gcd(m,n); \\ Michel Marcus, Mar 07 2015
A143665
a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals n.
Original entry on oeis.org
2, 3, 31, 13, 11, 7, 19531, 313, 19, 521, 12207031, 601, 305175781, 29, 181, 17, 409, 5167, 191, 41, 379, 23, 8971, 390001, 101, 5227, 109, 234750601, 59, 61, 1861, 2593, 199, 3061, 211, 37, 149, 761, 79, 241, 2238236249, 43, 1644512641, 89, 1171, 47
Offset: 1
Cf.
A112927 (base 2),
A143663 (base 3),
A112092 (base 4),
A143665 (base 5),
A379639 (base 6),
A379640 (base 7),
A379641 (base 8),
A379642 (base 9),
A007138 (base 10),
A379644 (base 11),
A252170 (base 12).
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p = 2; t = Table[0, {100}]; While[p < 3000000001, a = MultiplicativeOrder[5, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p]; p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)
A064080
Zsigmondy numbers for a = 4, b = 1: Zs(n, 4, 1) is the greatest divisor of 4^n - 1^n (A024036) that is relatively prime to 4^m - 1^m for all positive integers m < n.
Original entry on oeis.org
3, 5, 7, 17, 341, 13, 5461, 257, 1387, 41, 1398101, 241, 22369621, 3277, 49981, 65537, 5726623061, 4033, 91625968981, 61681, 1826203, 838861, 23456248059221, 65281, 1100586419201, 13421773, 22906579627, 15790321, 96076792050570581
Offset: 1
A064083
Zsigmondy numbers for a = 7, b = 1: Zs(n, 7, 1) is the greatest divisor of 7^n - 1^n (A024075) that is relatively prime to 7^m - 1^m for all positive integers m < n.
Original entry on oeis.org
6, 1, 19, 25, 2801, 43, 137257, 1201, 39331, 2101, 329554457, 2353, 16148168401, 102943, 4956001, 2882401, 38771752331201, 117307, 1899815864228857, 1129901, 11898664849, 247165843, 4561457890013486057, 5762401, 79797014141614001
Offset: 1
A064079
Zsigmondy numbers for a = 3, b = 1: Zs(n, 3, 1) is the greatest divisor of 3^n - 1^n (A024023) that is relatively prime to 3^m - 1^m for all positive integers m < n.
Original entry on oeis.org
2, 1, 13, 5, 121, 7, 1093, 41, 757, 61, 88573, 73, 797161, 547, 4561, 3281, 64570081, 703, 581130733, 1181, 368089, 44287, 47071589413, 6481, 3501192601, 398581, 387440173, 478297, 34315188682441, 8401, 308836698141973, 21523361
Offset: 1
A109347
Zsigmondy numbers for a = 5, b = 3: Zs(n, 5, 3) is the greatest divisor of 5^n - 3^n (A005058) that is relatively prime to 5^m - 3^m for all positive integers m < n.
Original entry on oeis.org
2, 1, 49, 17, 1441, 19, 37969, 353, 19729, 421, 24325489, 481, 609554401, 10039, 216001, 198593, 381405156481, 12979, 9536162033329, 288961, 18306583, 6125659, 5960417405949649, 346561, 103408180634401, 152787181, 3853528045489, 179655841, 93132223146359169121
Offset: 1
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rad(n) = factorback(factor(n)[, 1])
lista(nn) = {prad = 1; for (n=1, nn, val = 5^n-3^n; d = divisors(val); gd = 1; forstep(k=#d, 1, -1, if (gcd(d[k], prad) == 1, g = d[k]; break)); print1(g, ", "); prad = ra(prad*val););} \\ Michel Marcus, Nov 15 2016
A064082
Zsigmondy numbers for a = 6, b = 1: Zs(n, 6, 1) is the greatest divisor of 6^n - 1^n (A024062) that is relatively prime to 6^m - 1^m for all positive integers m < n.
Original entry on oeis.org
5, 7, 43, 37, 311, 31, 55987, 1297, 46873, 1111, 72559411, 1261, 2612138803, 5713, 1406371, 1679617, 3385331888947, 46441, 121871948002099, 1634221, 1822428931, 51828151, 157946044610720563, 1678321, 731325737104301
Offset: 1
A323748
Square array read by ascending antidiagonals: the n-th row lists the Zsigmondy numbers for a = n, b = 1, that is, T(n,k) = Zs(k, n, 1) is the greatest divisor of n^k - 1 that is coprime to n^m - 1 for all positive integers m < k, with n >= 2, k >= 1.
Original entry on oeis.org
1, 2, 3, 3, 1, 7, 4, 5, 13, 5, 5, 3, 7, 5, 31, 6, 7, 31, 17, 121, 1, 7, 1, 43, 13, 341, 7, 127, 8, 9, 19, 37, 781, 13, 1093, 17, 9, 5, 73, 25, 311, 7, 5461, 41, 73, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 3, 37, 41, 4681, 43, 55987, 313, 1387, 61, 2047, 12, 13, 133, 101, 7381, 19, 137257, 1297, 15751, 41, 88573, 13
Offset: 2
In the following list, "*" identifies a prime power.
Table begins
n\k | 1 2 3 4 5 6 7 8
2 | 1 , 3*, 7*, 5*, 31*, 1 , 127*, 17*
3 | 2*, 1 , 13*, 5*, 121*, 7*, 1093*, 41*
4 | 3*, 5*, 7*, 17*, 341 , 13*, 5461 , 257*
5 | 4*, 3*, 31*, 13*, 781 , 7*, 19531*, 313*
6 | 5*, 7*, 43*, 37*, 311*, 31*, 55987*, 1297*
7 | 6 , 1 , 19*, 25*, 2801*, 43*, 137257 , 1201*
8 | 7*, 9*, 73*, 65 , 4681 , 19*, 42799 , 4097
9 | 8*, 5*, 91 , 41*, 7381 , 73*, 597871 , 3281
10 | 9*, 11*, 37*, 101*, 11111 , 91 , 1111111 , 10001
11 | 10 , 3*, 133 , 61*, 3221*, 37*, 1948717 , 7321*
12 | 11*, 13*, 157*, 145 , 22621*, 133 , 3257437 , 20737
The first few columns:
T(n,1) = n - 1;
T(n,2) = A000265(n+1);
T(n,3) = (n^2 + n + 1)/3 if n == 1 (mod 3), n^2 + n + 1 otherwise;
T(n,4) = (n^2 + 1)/2 if n == 1 (mod 2), n^2 + 1 otherwise;
T(n,5) = (n^4 + n^3 + n^2 + n + 1)/5 if n == 1 (mod 5), n^4 + n^3 + n^2 + n + 1 otherwise;
T(n,6) = (n^2 - n + 1)/3 if n == 2 (mod 3), n^2 - n + 1 otherwise;
T(n,7) = (n^6 + n^5 + ... + 1)/7 if n == 1 (mod 7), n^6 + n^5 + ... + 1 otherwise;
T(n,8) = (n^4 + 1)/2 if n == 1 (mod 2), n^4 + 1 otherwise;
T(n,9) = (n^6 + n^3 + 1)/3 if n == 1 (mod 3), n^6 + n^3 + 1 otherwise;
T(n,10) = (n^4 - n^3 + n^2 - n + 1)/5 if n == 4 (mod 5), n^4 - n^3 + n^2 - n + 1 otherwise;
T(n,11) = (n^10 + n^9 + ... + 1)/11 if n == 1 (mod 11), n^10 + n^9 + ... + 1 otherwise;
T(n,12) = n^4 - n^2 + 1 (12 is not of the form p^e*d for any prime p, exponent e >= 1 and d dividing p-1).
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Table[Function[n, SelectFirst[Reverse@ Divisors[n^k - 1], Function[m, AllTrue[n^Range[k - 1] - 1, GCD[#, m] == 1 &]]]][j - k + 2], {j, 12}, {k, j}] // Flatten (* or *)
Table[Function[n, If[k == 2, #/2^IntegerExponent[#, 2] &[n + 1], #/GCD[#, k] &@ Cyclotomic[k, n]]][j - k + 1], {j, 2, 13}, {k, j - 1}] // Flatten (* Michael De Vlieger, Feb 02 2019 *)
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T(n,k) = if(k==2, (n+1)>>valuation(n+1, 2), my(m = polcyclo(k, n)); m/gcd(m, k))
A093109
Numbers n such that the Zsigmondy number Zs(n,5,1) differs from the n-th cyclotomic polynomial evaluated at 5.
Original entry on oeis.org
2, 4, 6, 8, 16, 18, 32, 42, 52, 54, 55, 64, 93, 128, 162, 171, 256, 272, 294, 355, 406, 486, 506, 512, 605, 676, 820, 1024, 1332, 1458, 1474, 1711, 1806, 1830, 2048, 2058, 2162, 2504, 2525, 2715, 2756, 2883, 2943, 3081, 3249, 3629, 3916, 4096, 4374, 4624, 5210
Offset: 1
Showing 1-9 of 9 results.
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