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.
A094465
Least initial value for an Euclid/Mullin sequence whose 4th term is prime(n). prime(1)=2 is never a fourth term, so offset=2.
Original entry on oeis.org
5, 19, 43, 31, 67, 541, 193, 157, 1213, 811, 487, 2371, 2, 1543, 733, 1319, 1291, 1753, 1621, 2713, 13, 1231, 2833, 2053, 1801, 3313, 5011, 821, 2467, 5101, 3253, 8573, 3637, 1553, 15427, 5521, 3191, 9173, 7237, 10531, 11071, 6271, 9103, 15727, 7993
Offset: 2
n=14: prime(14) = 43 and an Euclid-Mullin sequence started with a(14) = 2 = prime(1) is {2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...} is A000945, the prototype EM-sequence.
n=7: a(7) = prime(100) = 541, with EM sequence as follows: {541, 2, 3, 17, 139, 7, 1871, 100457892907, 19, 11047, ...}, where the 4th term equals prime(n) = prime(7) = 17.
It is characteristic but not so simple congruence relations holds of term(1) mod term(4) form for various first or 4th primes, not necessarily smallest ones. See comment at A094464.
-
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=4}; ta[[j]]=a[el], {j, 1, 20000}] Table[Prime[Min[Flatten[Position[ta, Prime[w]]]]], {w, 1, 100}]
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.
Comments