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.

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A257374 Numbers n such that n, n+4, n+10, n+12, n+16, n+22, n+24, n+30, n+36, n+40, n+42, n+46, n+52, n+54, n+60, n+64 and n+66 are all prime.

Original entry on oeis.org

734975534793324512717947, 753314125249587933791677, 1341829940444122313597407
Offset: 1

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Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of this sequence, A257375, A257376, A257377.

Extensions

a(3) from Norman Luhn, Oct 27 2021

A257375 Numbers n such that n, n+4, n+6, n+10, n+16, n+18, n+24, n+28, n+30, n+34, n+40, n+46, n+48, n+54, n+58, n+60 and n+66 are all prime.

Original entry on oeis.org

13, 47624415490498763963983, 78314167738064529047713, 83405687980406998933663, 110885131130067570042703, 163027495131423420474913
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, this sequence, A257376, A257377.

A257376 Numbers n such that n, n+6, n+8, n+12, n+18, n+20, n+26, n+32, n+36, n+38, n+42, n+48, n+50, n+56, n+60, n+62 and n+66 are all prime.

Original entry on oeis.org

1620784518619319025971, 2639154464612254121531, 3259125690557440336631, 124211857692162527019731
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, this sequence, A257377.

Extensions

a(1) corrected by Tim Johannes Ohrtmann, Dec 17 2015

A257377 Numbers n such that n, n+2, n+6, n+12, n+14, n+20, n+24, n+26, n+30, n+36, n+42, n+44, n+50, n+54, n+56, n+62 and n+66 are all prime.

Original entry on oeis.org

17, 37630850994954402655487, 53947453971035573715707, 174856263959258260646207, 176964638100452596444067, 207068890313310815346497, 247620555224812786876877, 322237784423505559739147
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, this sequence.

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

A098420 Members of prime triples (p,q,r) with p < q < r = p + 6.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 37, 41, 43, 47, 67, 71, 73, 97, 101, 103, 107, 109, 113, 191, 193, 197, 199, 223, 227, 229, 233, 277, 281, 283, 307, 311, 313, 317, 347, 349, 353, 457, 461, 463, 467, 613, 617, 619, 641, 643, 647, 821, 823, 827, 829, 853, 857, 859, 863
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 07 2004

Keywords

Comments

A098418(a(n)) > 0; complement of A098419 in A000040.
Union of A007529, A098414 and A098415.

Links

Crossrefs

Cf. A000040, A007529, A098414, A098415, A098418, A098419.

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p2=p+2]&&PrimeQ[p6=p+6], AppendTo[lst, p];AppendTo[lst, p2];AppendTo[lst, p6]];If[PrimeQ[p4=p+4]&&PrimeQ[p6=p+6], AppendTo[lst, p];AppendTo[lst, p4];AppendTo[lst, p6]], {n, 6!}];Union[lst] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)

A098418 Number of prime triples (p,q,r) with p

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 3, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 07 2004

Keywords

Comments

0 <= a(n) <= 3;
a(A098419(n))=0; a(A098420(n))>0; a(A098421(n))=1; a(A098422(n))=2; a(A098423(n))=3.

Examples

			A000040(13)=41: A007529(7)=41, A098414(6)=41 and
A098415(k)<>41 for all k, therefore a(13)=2.

Links

Crossrefs

Cf. A007529, A098414, A098415.

A098424 Number of prime triples (p,q,r) <= n with p

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 07 2004

Keywords

Comments

Convention: a prime triple is <= n iff its smallest member is <= n;
a(n) <= A098428(n).

Examples

			a(15) = #{(5,7,11),(7,11,13),(11,13,17),(13,17,19)} = 4.

Links

Crossrefs

Cf. A007529, A098414, A098415, A071538.
Cf. A010051, A000040.

Programs

  • Haskell
    a098424 n = length [(p,q,r) | p <- takeWhile (<= n) a000040_list,
            let r = p + 6, a010051 r == 1, q <- [p+1..r-1], a010051 q == 1]
-- Reinhard Zumkeller, Nov 15 2011
  • Mathematica
    With[{pts=Select[Partition[Prime[Range[1200]],3,1],Last[#]-First[#] == 6&]}, Table[Count[pts,?(First[#]<=n&)],{n,110}]] (* _Harvey P. Dale, Nov 09 2011 *)

A098423 Primes occurring in exactly three prime triples (p,q,r) with p

Original entry on oeis.org

11, 13, 17, 103, 107, 1487, 1873, 3463, 5653, 15733, 16063, 16067, 19423, 19427, 21017, 22277, 43783, 43787, 55337, 79693, 88813, 101113, 144167, 165707, 166847, 195737, 201827, 225347, 247607, 257863, 266683, 268817, 276043, 284743
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 07 2004

Keywords

Comments

A098418(a(n)) = 3; subsequence of A098420.
This sequence consists of all integers of the form (prime(m)*prime(m+4)+36)/prime(m+2), for m>0, where prime(m) = A000040(m). Also note that the integers resulting from that rule equal prime(m+2), therefore a(n) also consists of all integers of the form sqrt[prime(m)*prime(m+4)+36]. - Richard R. Forberg, Jan 11 2016

Examples

			A000040(27)=103: A007529(11)=103, A098414(10)=103 and A098415(9)=103, therefore 103 is a term.

Links

Crossrefs

Cf. A098419, A098421, A098422.

A279765 Primes p such that p+24 and p+48 are also primes.

Original entry on oeis.org

5, 13, 19, 23, 59, 79, 83, 89, 103, 149, 233, 269, 283, 349, 373, 409, 419, 439, 443, 499, 523, 569, 593, 653, 709, 773, 829, 839, 859, 863, 929, 1039, 1069, 1259, 1279, 1399, 1423, 1559, 1699, 1753, 1823, 1949, 1979, 2039, 2063, 2089, 2113, 2309, 2333, 2393
Offset: 1

Views

Author

Gerhard Kirchner, Dec 18 2016

Keywords

Comments

Subsequence of A033560. The triples have the form (p,p+d,p+2d). The current sequence (d=24) continues A023241 (d=6), A185022 (d=12) and A156109 (d=18). The frequencies of such triples and the triple (p, p+3±1, p+6) in A007529 do not differ very much (see table in the link "comparison of triples"). For creating the b-file I used a file of prime differences, divided by 2 (extension of A028334). For filling the table I analyzed primes up to 10^9.
Annotation: The algorithm using a file of primes or prime differences is not difficult but not as easy as using a function like isprime(n). On the other hand, such a function needs computing time which is not negligible for large numbers.

Examples

			First term: 5, 5 + 24 = 29 and 5 + 48 = 53 are all primes.

Links

Crossrefs

Cf. A023241, A033560, A156109, A185022.

Programs

  • Mathematica
    Select[Prime@Range@500, PrimeQ[# + 24] && PrimeQ[# + 48] &] (* Robert G. Wilson v, Dec 18 2016 *)
  • PARI
    is(n) = for(k=0, 2, if(!ispseudoprime(n+24*k), return(0))); 1 \\ Felix Fröhlich, Dec 26 2016
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