A125045
Odd primes generated recursively: a(1) = 3, a(n) = Min {p is prime; p divides Q+2}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
3, 5, 17, 257, 65537, 641, 7, 318811, 19, 1747, 12791, 73, 90679, 67, 59, 113, 13, 41, 47, 151, 131, 1301297155768795368671, 20921, 1514878040967313829436066877903, 5514151389810781513, 283, 1063, 3027041, 29, 24040758847310589568111822987, 154351, 89
Offset: 1
a(7) = 7 is the smallest prime divisor of 3 * 5 * 17 * 257 * 65537 * 641 + 2 = 2753074036097 = 7 * 11 * 37 * 966329953.
-
a={3}; q=1;
For[n=2,n<=20,n++,
q=q*Last[a];
AppendTo[a,Min[FactorInteger[q+2][[All,1]]]];
];
a (* Robert Price, Jul 16 2015 *)
A124984
Primes of the form 8*k + 3 generated recursively. Initial prime is 3. General term is a(n) = Min_{p is prime; p divides 2 + Q^2; p == 3 (mod 8)}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
3, 11, 1091, 1296216011, 2177870960662059587828905091, 76870667, 19, 257680660619, 73677606898727076965233531, 23842300525435506904690028531941969449780447746432390747, 35164737203
Offset: 1
a(3) = 1091 is the smallest prime divisor congruent to 3 mod 8 of 2+Q^2 = 1091, where Q = 3 * 11.
- D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.
-
a = {3}; q = 1;
For[n = 2, n ≤ 5, n++,
q = q*Last[a];
AppendTo[a, Min[Select[FactorInteger[2 + q^2][[All, 1]], Mod[#,
8] \[Equal] 3 &]]];
];
a (* Robert Price, Jul 14 2015 *)
-
lista(nn) = my(f, q=3); print1(q); for(n=2, nn, f=factor(2+q^2)[, 1]~; for(i=1, #f, if(f[i]%8==3, print1(", ", f[i]); q*=f[i]; break))); \\ Jinyuan Wang, Aug 05 2022
A125037
Primes of the form 26k+1 generated recursively. Initial prime is 53. General term is a(n) = Min {p is prime; p divides (R^13 - 1)/(R - 1); p == 1 (mod 13)}, where Q is the product of previous terms in the sequence and R = 13*Q.
Original entry on oeis.org
53, 11462027512399586179504472990060461, 25793, 178907, 131, 5669, 3511, 157, 59021, 13070705295701, 547, 79, 424361132339, 126146525792794964042953901, 5889547, 521, 1301, 6249393047, 9829, 2549, 298378081, 29379481, 56993, 1093, 26729
Offset: 1
a(2) = 11462027512399586179504472990060461 is the smallest prime divisor congruent to 1 mod 26 of (R^13 - 1)/(R - 1) = 11462027512399586179504472990060461, where Q = 53 and R = 13*Q.
- M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.
-
a={53}; q=1;
For[n=2,n<=5,n++,
q=q*Last[a]; r=13*q;
AppendTo[a,Min[Select[FactorInteger[(r^13-1)/(r-1)][[All,1]],Mod[#,26]==1 &]]];
];
a (* Robert Price, Jul 16 2015 *)
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.
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 *)
Showing 1-10 of 17 results.
Comments