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 *)
A125038
Primes of the form 34k+1 generated recursively. Initial prime is 103. General term is a(n) = Min {p is prime; p divides (R^17 - 1)/(R - 1); p == 1 (mod 17)}, where Q is the product of previous terms in the sequence and R = 17*Q.
Original entry on oeis.org
103, 307, 9929, 187095201191, 76943, 37061, 137, 5615258941637, 302125531, 18089, 613, 409, 9419, 193189
Offset: 1
a(2) = 307 is the smallest prime divisor congruent to 1 mod 34 of (R^17 - 1)/(R-1) = 7813154903878257490980895975711871949096304270238017 = 307 * 326669135226428664734261 * 77907623430368753779713071, where Q = 103 and R = 17*Q.
- M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.
-
a={103}; q=1;
For[n=2,n<=5,n++,
q=q*Last[a]; r=17*q;
AppendTo[a,Min[Select[FactorInteger[(r^17-1)/(r-1)][[All,1]],Mod[#,34]==1 &]]];
];
a (* Robert Price, Jul 14 2015 *)
A125040
Primes of the form 16k+1 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^8 + 1}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
17, 47441, 5136468762577, 1217, 2413992194819190142614641, 113, 52654897, 241, 5310928841473, 673
Offset: 1
a(3) = 5136468762577 is the smallest prime divisor of (2Q)^8 + 1 = 45820731194492299767895461612240999140120699535617 = 5136468762577 * 33000748370307713 * 270317134666005456817, where Q = 17 * 47441.
- G. A. Jones and J. M. Jones, Elementary Number Theory, Springer-Verlag, NY, (1998), p. 271.
-
a = {17}; q = 1;
For[n = 2, n <= 3, n++,
q = q*Last[a];
AppendTo[a, Min[Select[FactorInteger[(2*q)^8 + 1][[All, 1]],
Mod[#, 16] == 1 &]]];
];
a (* Robert Price, Jul 14 2015 *)
A125041
Primes of the form 24k+17 generated recursively. Initial prime is 17. General term is a(n) = Min {p is prime; p divides (2Q)^4 + 1; p == 17 (mod 24)}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
17, 1336337, 4261668267710686591310687815697, 41, 904641301321079897900944986453955254215268639579197293450763646548520041534444726724543203327659858344185865089, 3449, 18701609, 8009, 38599161306788868932168755721, 857, 130073, 1433, 113, 809, 18954775793
Offset: 1
a(3) = 4261668267710686591310687815697 is the smallest prime divisor congruent to 17 mod 24 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.
A124985
Primes of the form 8*k + 7 generated recursively. Initial prime is 7. General term is a(n) = Min_{p is prime; p divides 8*Q^2 - 1; p == 7 (mod 8)}, where Q is the product of the previous terms.
Original entry on oeis.org
7, 23, 207367, 1902391, 167, 1511, 28031, 79, 3142977463, 2473230126937097422987916357409859838765327, 2499581669222318172005765848188928913768594409919797075052820591, 223
Offset: 1
a(4) = 1902391 is the smallest prime divisor, congruent to 7 modulo 8, of 8*Q^2 - 1 = 8917046441372551 = 97 * 1902391 * 48322513, where Q = 7 * 23 * 207367.
- D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 182.
-
a={7}; q=1;
For[n=2,n<=9,n++,
q=q*Last[a];
AppendTo[a,Min[Select[FactorInteger[8*q^2-1][[All,1]],Mod[#,8]==7&]]];
];
a (* Robert Price, Jul 18 2015 *)
-
main(size)={my(v=vector(size),i,q=1,t);for(i=1,size,t=1;while(!(prime(t)%8==7&&(8*q^2-1)%prime(t)==0),t++);v[i]=prime(t);q*=v[i]);v;} /* Anders Hellström, Jul 18 2015 */
A124992
Primes of the form 14k+1 generated recursively. Initial prime is 29. General term is a(n)=Min {p is prime; p divides (R^7 - 1)/(R - 1); Mod[p,7]=1}, where Q is the product of previous terms in the sequence and R = 7Q.
Original entry on oeis.org
29, 70326806362093, 43, 127, 59221, 113, 32411, 71, 4957, 74509, 4271, 19013, 239, 2003, 463, 421, 613575503674084673, 32089, 211, 54601, 3109
Offset: 1
a(3) = 43 is the smallest prime divisor congruent to 1 mod 14 of (R^7 - 1)/(R-1) =
8466454975669959912248567627122565866080343755024168315838344565727361366925647440393797835238961
= 43 * 10781 * 391441 * 428597443 * 11795628769 * 408944901028399 * 22566921596365593811470735460776534824496318810581339, where Q = 29 * 70326806362093 and R = 7Q.
- M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.
-
a={29}; q=1;
For[n=2,n<=3,n++,
q=q*Last[a]; r=7*q;
AppendTo[a,Min[Select[FactorInteger[(r^7-1)/(r-1)][[All,1]],Mod[#,14]==1 &]]];
];
a (* Robert Price, Jul 14 2015 *)
A125042
Primes of the form 48k+17 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^8 + 1; Mod[p,48]=17}, where Q is the product of previous terms in the sequence.
Original entry on oeis.org
17, 47441, 33000748370307713, 21377
Offset: 1
a(3) = 33000748370307713 is the smallest prime divisor congruent to 17 mod 48 of (2Q)^8 + 1 = 45820731194492299767895461612240999140120699535617 = 5136468762577 * 33000748370307713 * 270317134666005456817, where Q = 17 * 47441.
- G. A. Jones and J. M. Jones, Elementary Number Theory, Springer-Verlag, NY, (1998), p. 271.
-
a = {17}; q = 1;
For[n = 2, n ≤ 2, n++,
q = q*Last[a];
AppendTo[a, Min[Select[FactorInteger[(2*q)^8 + 1][[All, 1]],
Mod[#, 48] \[Equal] 17 &]]];
];
a (* Robert Price, Jul 14 2015 *)
A125043
Primes of the form 18k+1 generated recursively. Initial prime is 19. General term is a(n) = Min {p is prime; p divides (R^9 - 1)/(R^3 - 1); p == 1 (mod 9)}, where Q is the product of previous terms in the sequence and R = 3*Q.
Original entry on oeis.org
19, 20593, 163, 8321800321246060993879, 9002496685879, 9736549840211105800055992105260095004185761, 1117, 48871, 37, 109, 2072647, 811, 2647, 22934467, 73, 10715232331, 4861, 127, 883, 699733, 19918378819555761579853986597710971
Offset: 1
a(3) = 163 is the smallest prime divisor congruent to 1 mod 18 of (R^9-1)/(R^3-1) = 2615573032645879161713714169238484203 = 163 * 88080931 * 161773561 * 1126133310262611691, where Q = 19 * 20593 and R = 3*Q.
- M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), p. 209.
A125044
Primes of the form 54k+1 generated recursively. Initial prime is 109. General term is a(n) = Min {p is prime; p divides (R^27 - 1)/(R^9 - 1); p == 1 (mod 27)}, where Q is the product of previous terms in the sequence and R = 3*Q.
Original entry on oeis.org
109, 50221, 379, 5077, 2527181639419400128997560106426867837203, 112807, 2094067, 1567, 9325207, 370603, 67447, 27978113462777647321591, 1012771, 163, 396577, 7096357, 3511, 3673, 541, 389287, 1999, 68979565009, 649108891
Offset: 1
a(2) = 50221 is the smallest prime divisor congruent to 1 mod 54 of
(R^27-1)/(R^9- 1) = 1827509098737085519727094436535854935801097657 = 50221 * 106219 * 342587871163695447795790279515751543, where Q = 109 and R = 3*Q.
- M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), p. 209.
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