A124988
Primes of the form 12k+7 generated recursively. Initial prime is 7. General term is a(n)=Min {p is prime; p divides 3+4Q^2; Mod[p,12]=7}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
7, 199, 7761799, 487, 67, 103, 1482549740515442455520791, 31, 139, 787, 19, 39266047, 1955959, 50650885759, 367, 185767, 62168707
Offset: 1
a(3) = 1482549740515442455520791 is the smallest prime divisor congruent to 7 mod 12 of 3+4Q^2 = 5281642303363312989311974746340327 = 3562539697 * 1482549740515442455520791, where Q = 7 * 199 * 7761799 * 487 * 67 * 103.
-
a={7}; q=1;
For[n=2,n<=7,n++,
q=q*Last[a];
AppendTo[a,Min[Select[FactorInteger[4*q^2+3][[All,1]],Mod[#,12]==7 &]]];
];
a (* Robert Price, Jul 15 2015 *)
A124989
Primes of the form 10*k + 9 generated recursively. Initial prime is 19. General term is a(n) = Min_{p is prime; p divides 100*Q^2 - 5; p == 9 (mod 10)}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
19, 7219, 462739, 509, 129229, 295380580489, 9653956849, 149, 110212292237172705230749846071050188009093377022084806290042881946231583507557298889, 157881589, 60397967745386189, 1429, 79
Offset: 1
a(3) = 462739 is the smallest prime divisor congruent to 9 mod 10 of 100Q^2-5 = 1881313992095 = 5 * 462739 * 813121, where Q = 19 * 7219.
-
a={19}; q=1;
For[n=2,n<=6,n++,
q=q*Last[a];
AppendTo[a,Min[Select[FactorInteger[100*q^2-5][[All,1]],Mod[#,10]==9&]]];
];
a (* Robert Price, Jul 18 2015 *)
A124990
Primes of the form 12k+1 generated recursively. Initial prime is 13. General term is a(n)=Min {p is prime; p divides Q^4-Q^2+1}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
13, 28393, 128758492789, 73, 193, 37, 457, 8363172060732903211423577787181
Offset: 1
a(3) = 128758492789 is the smallest prime divisor of Q^4 - Q^2 + 1 = 18561733755472408508281 = 128758492789 * 144159296629, where Q = 13 * 28393.
- K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, NY, Second Edition (1990), p. 63.
-
a = {13}; q = 1;
For[n = 2, n ≤ 8, n++,
q = q*Last[a];
AppendTo[a, Min[Select[FactorInteger[q^4 - q^2 + 1][[All, 1]],
Mod[#, 12] == 1 &]]];
];
a (* Robert Price, Jun 25 2015 *)
A125039
Primes of the form 8k+1 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^4 + 1}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
17, 1336337, 4261668267710686591310687815697, 41, 4390937134822286389262585915435960722186022220433, 241, 1553, 243537789182873, 97, 27673, 4289, 457, 137201, 73, 337, 569891669978849, 617, 1697, 65089, 1609, 761
Offset: 1
a(3) = 4261668267710686591310687815697 is the smallest prime divisor of (2Q)^4 + 1 = 4261668267710686591310687815697, where Q = 17 * 1336337.
- G. A. Jones and J. M. Jones, Elementary Number Theory, Springer-Verlag, NY, (1998), p. 271.
A217759
Primes of the form 4k+3 generated recursively: a(1)=3, a(n)= Min{p; p is prime; Mod[p,4]=3; p|4Q^2-1}, where Q is the product of all previous terms in the sequence.
Original entry on oeis.org
3, 7, 43, 19, 6863, 883, 23, 191, 2927, 205677423255820459, 11, 163, 227, 9127, 59, 31, 71, 131627, 2101324929412613521964366263134760336303, 127, 1302443, 4065403, 107, 2591, 21487, 223, 12823, 167, 53720906651, 5452254637117019, 39827899, 11719, 131
Offset: 1
a(10) is 205677423255820459 because it is the only prime factor congruent to 3 (mod 4) of 4*(3*7*43*19*6863*883*23*191*2927)^2-1 = 5*13*2088217*256085119729*205677423255820459. The four smaller factors are all congruent to 1 (mod 4).
- Dirichlet,P.G.L (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 13.
A057205
Primes congruent to 3 modulo 4 generated recursively: a(n) = Min_{p, prime; p mod 4 = 3; p|4Q-1}, where Q is the product of all previous terms in the sequence. The initial term is 3.
Original entry on oeis.org
3, 11, 131, 17291, 298995971, 8779, 594359, 59, 151, 983, 19, 38851089348584904271503421339, 2359886893253830912337243172544609142020402559023, 823818731, 2287, 7, 9680188101680097499940803368598534875039120224550520256994576755856639426217960921548886589841784188388581120523, 163, 83, 1471, 34211, 2350509754734287, 23567
Offset: 1
a(4) = 17291 = 4*4322 + 3 is the smallest prime divisor congruent to 3 (mod 4) of Q = 3*11*131 - 1 = 17291.
- P. G. L. Dirichlet (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.
-
a={3}; q=1;
For[n=2,n<=7,n++,
q=q*Last[a];
AppendTo[a,Min[Select[FactorInteger[4*q-1][[All,1]],Mod[#,4]==3&]]];
];
a (* Robert Price, Jul 18 2015 *)
A094464
Least initial value for a Euclid/Mullin sequence whose 3rd term (= least prime divisor of 1+2p) equals the n-th prime. prime(1)=2 is never a third term, so offset=2.
Original entry on oeis.org
7, 17, 2, 5, 149, 263, 389, 11, 449, 821, 1091, 881, 1913, 23, 2729, 29, 2531, 6599, 2591, 6971, 3989, 41, 4583, 5189, 7019, 7673, 53, 13679, 7853, 8699, 12641, 9521, 13691, 12143, 19403, 13109, 22901, 83, 18251, 89, 20543, 32183, 23063, 26693
Offset: 2
n=25: prime(25) = 97 and an Euclid-Mullin sequence started with a(25) = 5189 = 97*53 + 48 is {5189, 2, 97, 101, 3, 7, 167, 356568143863}.
All larger (prime) solutions with 97 as 3rd term have the form 97k + 48 form. However, not all primes of the form 97k + 48 result in Euclid-Mullin (EM) sequences with the property that the 3rd term is 97. For example, 727 = 7*97 + 48 is a prime providing an EM sequence as follows: {727, 2, 3, 4363, 19, 5, 1709, 11, 33988283132431, 7} with 3rd term = 3.
Analogous statements hold for other initial or 3rd primes.
-
a[x_]:=First[Flatten[FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]]; ta=Table[0, {20000}];a[1]=1;Do[{a[1]=Prime[j], el=3}; ta[[j]]=a[el], {j, 1, 20000}] Table[Prime[Min[Flatten[Position[ta, Prime[w]]]]], {w, 1, 100}]
A124986
Primes of the form 12*k + 5 generated recursively. Initial prime is 5. General term is a(n) = Min_{p is prime; p divides 1 + 4*Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
5, 101, 1020101, 53, 29, 2507707213238852620996901, 449, 433361, 401, 925177698346131180901394980203075088053316845914981, 44876921, 17, 173
Offset: 1
a(8) = 433361 is the smallest prime divisor congruent to 5 mod 12 of 1+4Q^2 = 3179238942812523869898723304484664524974766291591037769022962819805514576256901 = 13 * 433361 * 42408853 * 2272998442375593325550634821 * 5854291291251561948836681114631909089, where Q = 5 * 101 * 1020101 * 53 * 29 * 2507707213238852620996901 * 449.
-
a={5}; q=1;
For[n=2,n<=5,n++,
q=q*Last[a];
AppendTo[a,Min[Select[FactorInteger[4*q^2+1][[All,1]],Mod[#,12]==5 &]]];
];
a (* Robert Price, Jul 16 2015 *)
A124987
Primes of the form 12k+5 generated recursively. Initial prime is 5. General term is a(n) = Min {p is prime; p divides 4+Q^2; p == 5 (mod 12)}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
5, 29, 17, 6076229, 1289, 78067083126343039013, 521, 8606045503613, 15837917, 1873731749, 809, 137, 2237, 17729
Offset: 1
a(3) = 17 is the smallest prime divisor congruent to 5 mod 12 of 4+Q^2 = 21029 = 17 * 1237, where Q = 5 * 29.
-
a={5}; q=1;
For[n=2,n<=5,n++,
q=q*Last[a];
AppendTo[a,Min[Select[FactorInteger[q^2+4][[All,1]],Mod[#,12]==5 &]]];
];
a (* Robert Price, Jul 16 2015 *)
A124991
Primes of the form 10k+1 generated recursively. Initial prime is 11. General term is a(n)=Min {p is prime; p divides (R^5 - 1)/(R - 1); Mod[p,5]=1}, where Q is the product of previous terms in the sequence and R = 5Q.
Original entry on oeis.org
11, 211, 1031, 22741, 41, 15487770335331184216023237599647357572461782407557681, 311, 61, 55172461, 3541, 1381, 2851, 19841, 151, 9033671, 456802301, 1720715817015281, 19001, 71
Offset: 1
a(3) = 1031 is the smallest prime divisor congruent to 1 mod 10 of (R^5 - 1)/(R - 1) = 18139194759758381 = 1031 * 17593787351851, where Q = 11 * 211 and R = 5Q.
- M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.
-
a={11}; q=1;
For[n=2,n<=6,n++,
q=q*Last[a]; r=5*q;
AppendTo[a,Min[Select[FactorInteger[(r^5-1)/(r-1)][[All,1]],Mod[#,10]==1&]]];
];
a (* Robert Price, Jul 14 2015 *)
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