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 *)
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 *)
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 */
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