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 1-7 of 7 results.

A022005 Initial members of prime triples (p, p+4, p+6).

Original entry on oeis.org

7, 13, 37, 67, 97, 103, 193, 223, 277, 307, 457, 613, 823, 853, 877, 1087, 1297, 1423, 1447, 1483, 1663, 1693, 1783, 1867, 1873, 1993, 2083, 2137, 2377, 2683, 2707, 2797, 3163, 3253, 3457, 3463, 3847, 4153, 4513, 4783, 5227, 5413, 5437, 5647, 5653, 5737, 6547
Offset: 1

Views

Author

Warut Roonguthai

Keywords

Comments

Subsequence of A029710. - R. J. Mathar, May 06 2017
All terms are congruent to 1 (modulo 6). - Matt C. Anderson, May 22 2015

Links

Crossrefs

Subsequence of A029710 and of A002476.
Subsequence of A007529.
Cf. A001359, A073649, A098413.

Programs

A098415 Greatest members r of prime triples (p,q,r) with p

Original entry on oeis.org

11, 13, 17, 19, 23, 43, 47, 73, 103, 107, 109, 113, 197, 199, 229, 233, 283, 313, 317, 353, 463, 467, 619, 647, 827, 829, 859, 863, 883, 887, 1093, 1097, 1283, 1303, 1307, 1429, 1433, 1453, 1487, 1489, 1493, 1613, 1669, 1699, 1789, 1873, 1877, 1879, 1999
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 07 2004

Keywords

Comments

Union of A098412 and A098413;
a(n)=A007529(n)+6; either a(n)=A098414(n)+2 or a(n)=A098414(n)+4.

Links

Crossrefs

Cf. A098424, A098416, A098417.

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[350]],3,1],#[[3]]- #[[1]] == 6&]][[3]] (* Harvey P. Dale, Mar 17 2015 *)
  • PARI
    is(n)=isprime(n) && isprime(n-6) && (isprime(n-2) || isprime(n-4)) \\ Charles R Greathouse IV, Feb 23 2017

A098412 Greatest members p of prime triples (p-6, p-4, p).

Original entry on oeis.org

11, 17, 23, 47, 107, 113, 197, 233, 317, 353, 467, 647, 827, 863, 887, 1097, 1283, 1307, 1433, 1487, 1493, 1613, 1877, 2003, 2087, 2243, 2273, 2663, 2693, 3257, 3467, 3533, 3677, 3923, 4007, 4133, 4523, 4643, 4793, 4937, 4973, 5237, 5483, 5507, 5657, 6203
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 07 2004

Keywords

Comments

Subsequence of A046117; a(n) = A073648(n) + 4 = A022004(n) + 6.

Links

Crossrefs

Cf. A098413, A098415.

Programs

  • Magma
    [p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-4)]; // Vincenzo Librandi, Dec 26 2010
  • Maple
    K:=10^7: # to get all terms <= K.
for n from 1 by 2 to K do; if isprime(n-6) and isprime(n-4) and isprime(n) then print(n) else fi; od;  # Muniru A Asiru, Aug 06 2017
  • Mathematica
    Select[Table[Prime[n], {n, 1000}], PrimeQ[# - 4] && PrimeQ[# - 6] &] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&][[All,3]] (* Harvey P. Dale, Sep 23 2017 *)

A073649 Define the composite field of a prime q to be f(q) = (t,s) if p, q, r are three consecutive primes and q-p = t and r-q = s. This is a sequence of primes q with field (4,2).

Original entry on oeis.org

11, 17, 41, 71, 101, 107, 197, 227, 281, 311, 461, 617, 827, 857, 881, 1091, 1301, 1427, 1451, 1487, 1667, 1697, 1787, 1871, 1877, 1997, 2087, 2141, 2381, 2687, 2711, 2801, 3167, 3257, 3461, 3467, 3851, 4157, 4517, 4787, 5231, 5417, 5441, 5651, 5657
Offset: 1

Views

Author

Amarnath Murthy, Aug 09 2002

Keywords

Links

Crossrefs

Cf. A073648, A073650.
Cf. A098413, A098414.
Equals A022005 + 4.

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[1200]],3,1],Differences[#] == {4,2}&]] [[2]] (* Harvey P. Dale, Jul 23 2011 *)

Extensions

Corrected and extended by Benoit Cloitre, Aug 13 2002

A144840 Numbers k such that the three numbers k-1, k+3 and k+5 are all prime.

Original entry on oeis.org

8, 14, 38, 68, 98, 104, 194, 224, 278, 308, 458, 614, 824, 854, 878, 1088, 1298, 1424, 1448, 1484, 1664, 1694, 1784, 1868, 1874, 1994, 2084, 2138, 2378, 2684, 2708, 2798, 3164, 3254, 3458, 3464, 3848, 4154, 4514, 4784, 5228, 5414, 5438, 5648, 5654, 5738
Offset: 1

Views

Author

Giovanni Teofilatto, Sep 23 2008

Keywords

Links

Crossrefs

Cf. A022005, A086801, A144834.
Cf. A098413, A144842.

Programs

  • Python
    from sympy import isprime
def ok(n): return n > 4 and isprime(n-1) and isprime(n+3) and isprime(n+5)
print(list(filter(ok, range(5739)))) # Michael S. Branicky, Aug 14 2021

Formula

a(n) = A022005(n) + 1. - R. J. Mathar, Sep 24 2008

Extensions

Definition edited and extended by R. J. Mathar, Sep 24 2008

A244452 Primes p such that p^2-2 and p^2+4 are also prime (i.e., initial members of prime triples (p, p^2-2, p^2+4)).

Original entry on oeis.org

3, 5, 7, 13, 37, 47, 103, 233, 293, 313, 607, 677, 743, 1367, 1447, 2087, 2543, 3023, 3803, 3863, 4093, 4153, 4373, 4583, 4643, 4793, 4957, 5087, 5153, 5623, 5683, 5923, 6287, 7177, 7247, 7547, 7817, 8093, 8527, 9133, 9403
Offset: 1

Views

Author

Felix Fröhlich, Jun 28 2014

Keywords

Comments

Intersection of A062326 and A062324.

Examples

			3 is in the sequence since it is the first member of the triple (3, 3^2-2, 3^2+4) and the resulting values in the triple (3, 7, 13) are all prime.

Links

Crossrefs

Cf. A022004, A073648, A098413.

Programs

  • Mathematica
    Select[Prime[Range[1200]],AllTrue[#^2+{4,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 28 2018 *)
  • PARI
    forprime(p=2, 10^4, if(isprime(p^2-2) && isprime(p^2+4), print1(p, ", ")))

A290635 Greatest of 4 consecutive primes with consecutive gaps 6, 4, 2.

Original entry on oeis.org

43, 73, 283, 619, 1303, 1669, 1789, 1873, 1999, 2143, 2383, 2689, 2803, 4519, 5419, 5443, 5653, 7879, 9013, 11833, 13693, 14563, 17389, 18133, 18313, 20359, 21493, 22159, 24109, 27283, 32719, 35533, 36793, 37573, 41233, 41959, 42409, 42463, 44269, 47149, 50593, 55219, 55819, 55933
Offset: 1

Views

Author

Muniru A Asiru, Aug 08 2017

Keywords

Comments

All terms = {13, 19} mod 30.

Examples

			43 is a member of the sequence because 43 is the greatest of the 4 consecutive primes 31, 37, 41, 43 with consecutive gaps 6, 4, 2; that is, 37 - 31 = 6, 41 - 37 = 4, 43 - 41 = 2.

Links

Crossrefs

Subsequence of A006512 and A098413.
Cf. A022005, A078855.

Programs

  • GAP
    K:=2*10^5+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=Reversed([2,4,6]);;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2]]);;
P3:=List(Positions(P2,I),i->P[i+Length(I)]);
# More efficient
  • GAP
    Filtered(Set(Flat(List([13,19],j->List([1..2000],i->30*i+j)))),j->IsPrime(j) and IsPrime(j-12) and not IsPrime(j-10) and not IsPrime(j-8) and IsPrime(j-6) and not IsPrime(j-4) and IsPrime(j-2)); # Muniru A Asiru, Jul 03 2018
  • Maple
    for i from 1 to 10^5 do if ithprime(i+1)=ithprime(i)+6 and ithprime(i+2)=ithprime(i)+4 and ithprime(i+3)=ithprime(i)+2  then print(ithprime(i+3)); fi; od; # Corrected by Robert Israel, Jun 28 2018
# More efficient:
primes:= select(isprime,[seq(seq(30*i+j,j=[13,19]),i=1..10^4)]):
select(t -> isprime(t-2) and isprime(t-6) and isprime(t-12) and not isprime(t-8), primes); # Robert Israel, Jun 28 2018
  • Mathematica
    With[{s = Differences@ Prime@ Range[10^4]}, Prime[1 + SequencePosition[s, {6, 4, 2}][[All, -1]] ] ] (* Michael De Vlieger, Aug 16 2017 *)
Select[Partition[Prime[Range[6000]],4,1],Differences[#]=={6,4,2}&][[All,4]] (* Harvey P. Dale, Feb 13 2022 *)
  • PARI
    is(n) = if(!ispseudoprime(n), return(0), my(v=[n-2, n-6, n-12]); if(v[1]==precprime(n-1) && v[2]==precprime(v[1]-1) && v[3]==precprime(v[2]-1), return(1))); 0 \\ Felix Fröhlich, Aug 10 2017

Formula

a(n) = A078855(n) + 12.
Showing 1-7 of 7 results.

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