cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 39 results. Next

A057204 Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.

Original entry on oeis.org

7, 199, 7761799, 487, 67, 103, 3562539697, 7251847, 13, 127, 5115369871402405003, 31, 697830431171707, 151, 3061, 229, 193, 5393552285540920774057256555028583857599359699, 709, 397, 37, 61, 46168741, 3127279, 181, 122268541
Offset: 1

Views

Author

Labos Elemer, Oct 09 2000

Keywords

Comments

4*Q^2 + 3 always has a prime divisor congruent to 1 modulo 6.
If we start with the empty product Q=1 then it is not necessary to specify the initial prime. - Jens Kruse Andersen, Jun 30 2014

Examples

			a(4)=487 is the smallest prime divisor of 4*Q*Q + 3 = 10812186007, congruent to 1 (mod 6), where Q = 7*199*7761799.

References

  • 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.

Links

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002476, A057204-A057208.

Programs

  • Mathematica
    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[#,6]==1 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)
  • PARI
    Q=1;for(n=1,11,f=factor(4*Q^2+3);for(i=1,#f~,p=f[i,1];if(p%6==1,break));print1(p", ");Q*=p) \\ Jens Kruse Andersen, Jun 30 2014

Extensions

More terms from Nick Hobson, Nov 14 2006
More terms from Sean A. Irvine, Oct 23 2014

A057208 Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.

Original entry on oeis.org

5, 29, 1237, 32171803229, 829, 405565189, 14717, 39405395843265000967254638989319923697097319108505264560061, 282860648026692294583447078797184988636062145943222437, 53, 421, 13, 109, 4133, 6476791289161646286812333, 461, 34549, 453690033695798389561735541
Offset: 1

Views

Author

Labos Elemer, Oct 09 2000

Keywords

Examples

			a(3) = 1237 = 8*154 + 5 is the smallest suitable prime divisor of (5*29)*5*29 + 4 = 21029 = 17*1237. (Although 17 is the smallest prime divisor, 17 is not congruent to 5 modulo 8.)

References

  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.

Links

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A007521, A057204-A057208.

Programs

  • Mathematica
    a={5}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4+q^2][[All,1]],Mod[#,8]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)
  • PARI
    lista(nn) = {v = vector(nn); v[1] = 5; print1(v[1], ", "); for (n=2, nn, f = factor(4 + prod(k=1, n-1, v[k])^2); for (k=1, #f~, if (f[k, 1] % 8 == 5, v[n] = f[k,1]; break);); print1(v[n], ", "););} \\ Michel Marcus, Oct 27 2014

Extensions

More terms from Sean A. Irvine, Oct 26 2014

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

Views

Author

Nick Hobson, Nov 18 2006

Keywords

Comments

The first five terms comprise the known Fermat primes: A019434.

Examples

			a(7) = 7 is the smallest prime divisor of 3 * 5 * 17 * 257 * 65537 * 641 + 2 = 2753074036097 = 7 * 11 * 37 * 966329953.

Links

Crossrefs

Cf. A000945, A019434, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

Programs

  • Mathematica
    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

Views

Author

Nick Hobson, Nov 18 2006

Keywords

Comments

2+Q^2 always has a prime divisor congruent to 3 modulo 8.

Examples

			a(3) = 1091 is the smallest prime divisor congruent to 3 mod 8 of 2+Q^2 = 1091, where Q = 3 * 11.

References

  • D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.

Links

Crossrefs

Cf. A000945, A007520, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

a(10) from Robert Price, Jul 04 2015
a(11) from Robert Price, Jul 05 2015

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

Views

Author

Nick Hobson, Nov 18 2006

Keywords

Comments

All prime divisors of (R^13 - 1)/(R - 1) different from 13 are congruent to 1 modulo 26.

Examples

			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.

References

  • M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.

Crossrefs

Cf. A000945, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

Programs

  • Mathematica
    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 *)

Extensions

More terms from Sean A. Irvine, Jun 24 2011

A124993 Primes of the form 22k+1 generated recursively. Initial prime is 23. General term is a(n) = Min {p is prime; p divides (R^11 - 1)/(R - 1); p == 1 (mod 11)}, where Q is the product of previous terms in the sequence and R = 11*Q.

Original entry on oeis.org

23, 4847239, 2971, 3936923, 9461, 1453, 331, 81373909, 89, 920771904664817214817542307, 353, 401743, 17088192002665532981, 11617
Offset: 1

Views

Author

Nick Hobson, Nov 18 2006

Keywords

Comments

All prime divisors of (R^11 - 1)/(R - 1) different from 11 are congruent to 1 modulo 22.

Examples

			a(3) = 2971 is the smallest prime divisor congruent to 1 mod 22 of (R^11-1)/(R-1) =
7693953366218628230903493622259922359469805176129784863956847906415055607909988155588181877
= 2971 * 357405886421 * 914268562437006833738317047149 * 7925221522553970071463867283158786415606996703, where Q = 23 * 4847239, and R = 11*Q.

References

  • M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), pp. 208-209.

Crossrefs

Cf. A000945, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

Programs

  • Mathematica
    a={23}; q=1;
For[n=2,n<=2,n++,
    q=q*Last[a]; r=11*q;
    AppendTo[a,Min[Select[FactorInteger[(r^11-1)/(r-1)][[All,1]],Mod[#,11]==1 &]]];
    ];
a (* Robert Price, Jul 14 2015 *)

Extensions

More terms from Max Alekseyev, May 29 2009

A051614 4th term in Euclid-Mullin prime sequence started with n-th prime (cf. A000945).

Original entry on oeis.org

43, 43, 3, 43, 3, 79, 3, 5, 3, 3, 11, 223, 3, 7, 3, 3, 827, 367, 13, 3, 439, 5, 3, 3, 11, 5, 619, 3, 5, 3, 7, 3, 3, 5, 5, 907, 23, 11, 3, 3, 3, 1087, 3, 19, 3, 5, 7, 13, 3, 5, 3, 3, 1447, 3, 3, 3, 3767, 1627, 1663, 3, 1699, 3, 19, 5, 1879, 3, 1987, 7, 3, 5, 4943, 3, 2203, 2239, 5, 23
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

First term in Euclid-Mullin sequence is p (say), 2nd term (if p odd) is 2, 3rd term is A023592.

Examples

			E.g., (5,2,11,3), (89,2,179,3), (17,2,5,3), (2,3,7,43), (61,2,3,367).

Links

Crossrefs

Cf. A000945, A000040, A005384, A005385, A023592, A051308 - A051335.

Programs

  • Mathematica
    a[n_] := (Clear[f]; f[1] = Prime[n]; f[k_] := f[k] = FactorInteger[Product[f[i], {i, 1, k-1}]+1][[1, 1]]; f[4]); Table[a[n], {n, 1, 76}] (* Jean-François Alcover, Feb 05 2014 *)

A093778 Primes p used as initial values for Euclid-Mullin sequences (variant A000945) instead of 2, such that all provide {p,2,3,5,7,11,13,q,...} initial segments in which the first six primes occur from 2nd to 7th terms.

Original entry on oeis.org

99109, 159169, 189199, 399409, 459469, 609619, 669679, 699709, 819829, 1030039, 1090099, 1150159, 1270279, 1300309, 1390399, 1420429, 1810819, 1870879, 1930939, 1960969, 2021029, 2051059, 2141149, 2201209, 2261269, 2321329
Offset: 1

Views

Author

Labos Elemer, May 03 2004

Keywords

Examples

			Initial segments of Euclid-Mullin sequences provided by
a[33]=3132139, a[34] and a[35] initial values:
{3132139,2,3,5,7,11,13,94058134171}}
{3282289,2,3,5,7,11,13,59}},
{3372379,2,3,5,7,11,13,29}}

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777.

Programs

  • Mathematica
    b[x_] :=First[Flatten[FactorInteger[Apply[Times, Table[b[j], {j, 1, x - 1}]] +1]]];b[1] = 1; Do[b[1] = Prime[j], el=8; If[Equal[Table[b[w], {w, 2, 7}], {2, 3, 5, 7, 11, 13}], Print[{j, Table[b[w], {w, 1, el}]}]], {j, 100000, 1000000}]

A094460 a(n) is the third term in Euclid-Mullin (EM) prime sequence initiated with n-th prime.

Original entry on oeis.org

7, 7, 11, 3, 23, 3, 5, 3, 47, 59, 3, 3, 83, 3, 5, 107, 7, 3, 3, 11, 3, 3, 167, 179, 3, 7, 3, 5, 3, 227, 3, 263, 5, 3, 13, 3, 3, 3, 5, 347, 359, 3, 383, 3, 5, 3, 3, 3, 5, 3, 467, 479, 3, 503, 5, 17, 7, 3, 3, 563, 3, 587, 3, 7, 3, 5, 3, 3, 5, 3, 7, 719, 3, 3, 3, 13, 19, 3, 11, 3, 839, 3, 863
Offset: 1

Views

Author

Labos Elemer, May 06 2004

Keywords

Examples

			First term is p[n], 2nd equals 2; 3rd term is given here as largest p-divisor of 2p+1 [occasionally safe primes, A005385];
4th terms listed in A051614; further terms are in A094461-A094463.

Crossrefs

Cf. A000945, A005385, A051614-A051616, A051308 - A051335, A093778-A093782, A094152/A094153.
Except for first term [which is A000945(3)], the same as A023592.

Programs

  • Mathematica
    a[x_]:=First[Flatten [FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]]; ta=Table[0, {168}]; a[1]=1; Do[{a[1]=Prime[j], el=10}; Print[a[el]; ta[[j]]=a[el]; j++ ], {j, 1, 168}]; ta

Formula

a(n)= a(n-1)+ A008472(a(n-1)) - Ctibor O. Zizka, May 26 2008

A093782 a(n) is the smallest initial value (a prime) for the Euclid-Mullin (EM) sequence in which the p=5 prime emerges as n-th term, i.e., arises at the n-th position.

Original entry on oeis.org

5, 0, 17, 19, 127, 61, 2, 31, 97, 13, 23, 269, 53, 239, 181, 449, 541, 11, 953, 1741, 179, 1889, 823, 3209, 13619, 383, 6971, 10331, 45959, 13721
Offset: 1

Views

Author

Labos Elemer, May 04 2004

Keywords

Comments

The sequence is not monotonic and it seems that p=5 may arise at any position > 2. a(2)=0 means that 5 is never the 2nd term in an EM sequence of A000945-type because a(2)=2 or 3.
a(31)>=8581. [Sean A. Irvine, Oct 31 2011]

Examples

			The sequence for 17 is 17, 2, 5, ... where the 5 is at the third place, therefore a(3)=17.
For n=15 we have the sequence 181, 2, 3, 1087, 73, 7, 29, 151, 61, 98689, 11, 10929259909, 678859, 97, 5, ...
a(16) = 449 uses the sequence 449, 2, 29, 3, 7, 349, 190861819, 166273, 16091, 11, 3807491, 53, 17, 313, 23, 5, ...
The sequence for 11 is 11, 2, 23, 3, 7, 13, 10805892983887, 73, 6397, 19, 489407, 2753, 87491, 18618443, 5, ... with the 5 at the 18th place, so a(18)=11.

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777-A093781.

Extensions

Corrected by R. J. Mathar, Oct 06 2006
a(16) = 449 was conjectured by R. J. Mathar and confirmed by Don Reble, Oct 07 2006
a(19)-a(24) from David Wasserman, Apr 20 2007
a(25)-a(30) from Sean A. Irvine, Oct 30 2011
Showing 1-10 of 39 results. Next

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