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 1-4 of 4 results.

A282321 Lesser of twin primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 191, 281, 311, 431, 461, 521, 641, 821, 881, 1031, 1061, 1091, 1151, 1301, 1451, 1481, 1721, 1871, 1931, 2081, 2111, 2141, 2381, 2591, 2711, 2801, 3251, 3371, 3461, 3581, 3671, 3821, 3851, 4001, 4091, 4241, 4271, 4421, 4481, 4721, 4931, 5021
Offset: 1

Views

Author

Martin Renner, Feb 11 2017

Keywords

Comments

The union of [this sequence and A282322] is A132241.
The union of [{3, 5}, this sequence, A282323 and A060229] is the lesser of twin primes sequence A001359.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.

Links

Crossrefs

Subset of A001097, A001359, A132232, A132241 and A132247.
Cf. A006512, A232880, A232881, A232882, A282322, A282323, A060229, A060229, A282326.

Programs

  • Magma
    [p: p in PrimesUpTo(5000) | IsPrime(p+2) and p mod 30 eq 11 ]; // Vincenzo Librandi, Feb 12 2017
  • Maple
    a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)+2) and ithprime(i) mod 30 = 11 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;
  • PARI
    list(lim)=my(v=List(), p=2); forprime(q=3, lim+2, if(q-p==2 && q%30==13, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A132242 Twin primes congruent to {17, 19} mod 30.

Original entry on oeis.org

17, 19, 107, 109, 137, 139, 197, 199, 227, 229, 347, 349, 617, 619, 827, 829, 857, 859, 1277, 1279, 1427, 1429, 1487, 1489, 1607, 1609, 1667, 1669, 1697, 1699, 1787, 1789, 1877, 1879, 1997, 1999, 2027, 2029, 2087, 2089, 2237, 2239, 2267, 2269, 2657, 2659
Offset: 1

Views

Author

Omar E. Pol, Aug 15 2007

Keywords

Links

Crossrefs

Cf. A001097, A001359, A006512, A039949.
Cf. A132243, A132241.

Programs

  • Mathematica
    a[0] = 7; a[n_] := a[n] = a[n - 1] + 10; Flatten[Table[If[PrimeQ[a[n]] && PrimeQ[a[n] + 2], {a[n],a[n] + 2}, {}], {n, 0, 1000}]] (* Roger L. Bagula, May 04 2008 *)

Extensions

More terms from Roger L. Bagula, May 04 2008

A282322 Greater of twin primes congruent to 13 (mod 30).

Original entry on oeis.org

13, 43, 73, 103, 193, 283, 313, 433, 463, 523, 643, 823, 883, 1033, 1063, 1093, 1153, 1303, 1453, 1483, 1723, 1873, 1933, 2083, 2113, 2143, 2383, 2593, 2713, 2803, 3253, 3373, 3463, 3583, 3673, 3823, 3853, 4003, 4093, 4243, 4273, 4423, 4483, 4723, 4933, 5023, 5233, 5443, 5503, 5653, 5743
Offset: 1

Views

Author

Martin Renner, Feb 11 2017

Keywords

Comments

The union of [A282321 and this sequence] is A132241.
The union of [{5, 7}, this sequence, A282324 and A282326] is the greater of twin primes sequence A006512.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.
A181604 without the 3. [Proof: working mod 10 we see that each value here is in A181604. For the other direction: Except 3 all twin primes in A181604 are upper twin primes; they cannot be lower twin primes because the upper ones would be multiples of 5. The twin primes in A181604 could be == 3 (mod 30) or == 13 (mod 30) or == 23 (mod 30). The first case is excluded because they would be multiples of 3; the third case is excluded because the lower twin primes would be == 21 (mod 30) and also multiples of 3. So only the case == 13 (mod 30) remains.] - R. J. Mathar, Feb 14 2017
Number of terms < 10^k for k >= 1: 0, 3, 13, 67, 401, 2736, 19797, 146841, 1141217, 9137078, ..., . - Robert G. Wilson v, Jan 07 2018

Links

Crossrefs

Subset of A001097, A006512, A132233, A132241 and A132247.
Cf. A001359, A232880, A232881, A232882, A282321, A282323, A282324, A282326.

Programs

  • Maple
    a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)-2) and ithprime(i) mod 30 = 13 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;
  • Mathematica
    Select[13 + 30 Range[0, 200], PrimeQ[# - 2] && PrimeQ[#] &] (* Robert G. Wilson v, Jan 07 2018 *)
  • PARI
    list(lim)=my(v=List(), p=2); forprime(q=3, lim, if(q-p==2 && q%30==13, listput(v, q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A295340 Numbers congruent to 11 or 13 mod 15.

Original entry on oeis.org

11, 13, 26, 28, 41, 43, 56, 58, 71, 73, 86, 88, 101, 103, 116, 118, 131, 133, 146, 148, 161, 163, 176, 178, 191, 193, 206, 208, 221, 223, 236, 238, 251, 253, 266, 268, 281, 283, 296, 298, 311, 313, 326, 328, 341, 343, 356, 358, 371, 373, 386, 388, 401, 403, 416, 418, 431, 433
Offset: 1

Views

Author

Mikk Heidemaa, Nov 20 2017

Keywords

Comments

Includes every prime and twin prime (as pairs of consecutive primes) congruent to 11 or 13 mod 30.

Links

Crossrefs

Cf. A132238 (subsequence of primes), A132241 (subsequence of twin primes).

Programs

  • Magma
    [n: n in [1..500] | n mod 15 in [11, 13]]; // Vincenzo Librandi, Sep 06 2018
  • Mathematica
    ParallelMap[11 * Ceiling[#/2] + 2 * # - 2 &, Range@ 10^3]
CoefficientList[ Series[(2x^2 + 2x + 11)/((1 + x) (x - 1)^2), {x, 0, 60}], x] (* or *)
LinearRecurrence[{1, 1, -1}, {11, 13, 26}, 60] (* Robert G. Wilson v, Jan 09 2018 *)
Select[Range[500], MemberQ[{11, 13}, Mod[#, 15]] &] (* Vincenzo Librandi, Sep 06 2018 *)
11/2 * Mod[#, 2] + 15 * #/2 - 2 &/@ Range@ 500 (* Mikk Heidemaa, Sep 08 2018 *)
  • PARI
    Vec(x*(11 + 2*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Dec 07 2017
  • PARI
    a(n) = if(n%2, (15*n+7)/2, (15*n-4)/2); \\ Altug Alkan, Sep 06 2018
  • PARI
    a(n) = [11, -2][(n - 1)%2 + 1] + 15*(n \ 2) \\ David A. Corneth, Sep 06 2018

Formula

a(n) = (1/4)*(-1)^n*(3*(-1)^n*(10*n + 1) - 11) for n > 0.
From Colin Barker, Dec 07 2017: (Start)
G.f.: x*(11 + 2*x + 2*x^2) / ((1 - x)^2*(1 + x)).
a(n) = (15*n - 4) / 2 for n even.
a(n) = (15*n + 7) / 2 for n odd.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
(End)
a(n) = ceiling(15*n/2) + 5*(n mod 2) - 2 for n > 0. - Mikk Heidemaa, Sep 06 2018
a(n + 2) = a(n) + 15. - David A. Corneth, Sep 06 2018
a(n) = (11/2)*(n mod 2) + 15*n/2 - 2 for n > 0. - Mikk Heidemaa, Sep 08 2018
f(n) = 15*n - ((13*n + 17) mod 26) for n > 0 yields odd terms. - Mikk Heidemaa, Oct 28 2019
a(n) = 11*ceiling(1/2*n) + 2*n - 2 for n > 0. - Mikk Heidemaa, Nov 04 2019
E.g.f.: 2 + ((30*x + 3)*exp(x) - 11*exp(-x))/4. - David Lovler, Sep 08 2022

Extensions

Name simplified by David A. Corneth, Sep 06 2018
Showing 1-4 of 4 results.

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