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.

A022008 Initial member of prime sextuples (p, p+4, p+6, p+10, p+12, p+16).

Original entry on oeis.org

7, 97, 16057, 19417, 43777, 1091257, 1615837, 1954357, 2822707, 2839927, 3243337, 3400207, 6005887, 6503587, 7187767, 7641367, 8061997, 8741137, 10526557, 11086837, 11664547, 14520547, 14812867, 14834707, 14856757, 16025827, 16094707, 18916477, 19197247
Offset: 1

Views

Author

Warut Roonguthai

Keywords

Comments

Without the initial 7, this gives primes at which difference pattern X42424Y (X and Y >= 8) occurs in A001223. - Labos Elemer
Subsequence of A022007. - Zak Seidov, Nov 01 2011
From Jean-Christophe Hervé, Sep 27 2014: (Start)
The primes in a sextuple a(n), a(n)+4, a(n)+6, a(n)+10, a(n)+12, a(n)+16 are consecutive since a(n)+2, a(n)+8 and a(n)+14 cannot be prime (multiple of 3).
The prime sextuples starting at a(n) give the highest concentration of primes that can occur on an interval of 17 integers (apart intervals starting at p < 7). It is conjectured that there are infinitely many such sextuples.
For n > 1, the 3 odd integers preceding and the 3 odd integers following the sextuple are not prime: a(n)-2 == a(n)+18 == 0 (mod 5), a(n)-4 == a(n)+20 == 0 (mod 3), a(n)-6 == a(n)+22 == 0 (mod 7) and thus a(n) == 97 (mod 210 = 2*3*5*7). (End)
All terms are congruent to 7 (mod 30). - Zak Seidov, May 07 2017
All terms but the first one are congruent to 97 (mod 210). - M. F. Hasler, Jan 18 2022

Examples

			n=2: 97, 101, 103, 107, 109, 113 are consecutive primes, while 91, 93, 95 and 115, 117 and 119 are not (cf. 4th comment about the border of composites).

Links

Crossrefs

Cf. A022007.
Cf. A001223, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A047078.
Cf. A350826 (number of n-digit terms).

Programs

  • GAP
    P:=Filtered([1,3..2*10^7+1],IsPrime);;  I:=[4,2,4,2,4];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
A022008:=List(Positions(List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]),I),j->P[j]); # Muniru A Asiru, Sep 03 2017
  • Magma
    [p: p in PrimesUpTo(2*10^7) | IsPrime(p+4) and IsPrime(p+6) and IsPrime(p+10)and IsPrime(p+12) and IsPrime(p+16)]; // Vincenzo Librandi, Aug 23 2015
  • Maple
    for i from 1 to 2*10^5 do if [ithprime(i+1), ithprime(i+2), ithprime(i+3), ithprime(i+4), ithprime(i+5)] = [ithprime(i)+4,ithprime(i)+6,ithprime(i)+10,ithprime(i)+12,ithprime(i)+16] then print(ithprime(i)); fi; od; # Muniru A Asiru, Sep 03 2017
  • Mathematica
    lst = {}; Do[p = Prime[n]; If[PrimeQ[p+4] && PrimeQ[p+6] && PrimeQ[p+10] && PrimeQ[p+12] && PrimeQ[p+16], AppendTo[lst, p]], {n, 1000000}]; lst
Transpose[Select[Partition[Prime[Range[10^6]],6,1],Differences[#]=={4,2,4,2,4}&]][[1]] (* Harvey P. Dale, Mar 15 2015 *)
  • PARI
    p=2;q=3;r=5;s=7;t=11;forprime(u=13,1e9,if(u-p==16 && p%3==1, print1(p", "));p=q;q=r;r=s;s=t;t=u) \\ Charles R Greathouse IV, Mar 29 2013
  • PARI
    {next_A022008(p, L=Vec(p+1,5), m=210, r=Mod(97,m))=for(i=1,oo, L[i%5+1]+16==(p=nextprime(p+1))&&break; p%m>111 && until(r==p=nextprime((p+8)\/210*210+97),); L[i%5+1]=p); p-16} \\ M. F. Hasler, Jan 18 2022
  • Perl
    use ntheory ":all"; say for sieve_prime_cluster(1,1e8, 4,6,10,12,16); # Dana Jacobsen, Sep 30 2015

A350825 Number of prime 5-tuples with initial member (A086140) between 10^(n-1) and 10^n.

Original entry on oeis.org

2, 2, 1, 4, 12, 44, 256, 1062, 5838
Offset: 1

Views

Author

M. F. Hasler, Mar 01 2022

Keywords

Comments

"Between 10^(n-1) and 10^n" is equivalent to saying "with n digits".
For n = 1 and n = 2, the last term of the last 5-tuple in that range (cf EXAMPLE) has one digit more than the initial term.
Terms a(1)-a(9) computed from b-files a(1..10000) for A022006 and A022007.

Examples

			a(1) = 2 because there are just two single-digit primes to start a prime 5-tuple, namely 5 = A022006(1) and 7 = A022007(1).
a(2) = 2 because 11 = A022006(2) and 97 = A022007(2) are the only two two-digit primes to start a prime 5-tuple.
a(3) = 1 because there is only one three-digit prime to start a prime 5-tuple, namely 101 = A022006(3).
Then there are a(4) = 4 four-digit primes, 1481, 1867, 3457 and 5647, which start a prime 5-tuple.

Crossrefs

Cf. A086140 (initial members p of prime quintuplets), A022006, A022007 (idem, specifically for patterns (p, p+2, ...) resp. (p, p+4, ...)).
Cf. A350826, A350827, A350828: similar for sextuplets, septuplets and octuplets.

Programs

  • PARI
    (D(v)=v[^1]-v[^-1])( [setsearch(A086140, 10^n, 1) | n<-[0..9]] ) \\ where A086140 is a vector of at least 7221 terms of that sequence.

A350827 Number of prime septuplets (i.e.: 7-tuples) with initial member (A022009 or A022010) between 10^(n-1) and 10^n.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 5, 21, 70, 370, 1862, 9634
Offset: 1

Views

Author

M. F. Hasler, Mar 01 2022

Keywords

Comments

"Between 10^(n-1) and 10^n" is equivalent to saying "with n digits".
Up to 10^600 at least, the largest term of all prime septuplets (= set of 7 consecutive primes {p1, ..., p7} with minimal possible diameter p7 - p1 = 20) has the same number of digits as the smallest term. (*)
Terms a(1)-a(12) computed from b-files a(1..10^4) for A022009 and A022010.
(*) We checked that n = 1, 2, 3, 5, 17 and 18 are the only values below 600 with more than 2 primes in the interval [10^n - 20, 10^n + 20]. So the probability of finding a 7-tuple with diameter 20 in such an interval seems exceedingly small. - M. F. Hasler, Apr 12 2022

Examples

			a(1) = a(3) = 0 because there is no single-digit nor a 3-digit prime to start a prime septuplet.
a(2) = a(4) = a(5) = 1 because 11 = A022009(1), 5639 = A022010(1) and 88799 = A022010(2) are the only prime with 2, 4 resp. 5 digits to start a prime septuplet.
Then there are a(6) = 4 six-digit primes, 165701, 284729, 626609 and 855719, which start a prime septuplet.

Crossrefs

Cf. A022009, A022010: initial members p of prime septuplets (p, p+2, p+6, ...) resp. (p, p+2, p+8, ...).
Cf. A350825, A350826, A350828: similar for quintuplets, sextuplets and octuplets.

Programs

  • PARI
    apply( {A350827(n,v=vector(6),c)=forprime(p=10^(n-1),10^n, v[n=1+n%#v]+20==(v[n]=p) && c++);c}, [1..8]) \\ becomes slow for n > 8. - M. F. Hasler, Apr 12 2022

A350828 Number of prime octuplets with initial member (A065706) between 10^(n-1) and 10^n.

Original entry on oeis.org

0, 2, 0, 1, 1, 3, 3, 9, 28, 136, 541, 2936
Offset: 1

Views

Author

M. F. Hasler, Mar 01 2022

Keywords

Comments

"Between 10^(n-1) and 10^n" is equivalent to saying "with n (decimal) digits".
A prime octuplet is a sequence of 8 consecutive primes (p1, ..., p8) of minimal diameter p8 - p1 = 26.
Terms a(1)-a(12) computed from b-file a(1..18123) for A065706. Using Luhn's database, cf. LINKS, one can get 3 more terms.
So far, the last term of all the octuplets has the same number of digits as the initial term.

Examples

			a(1) = a(3) = 0 because there is no single-digit nor a 3-digit prime initial member of a prime octuplet.
a(2) = 2 because 11 and 17 are the only 2-digit members of A065706, i.e., primes to start a prime octuplet.
a(4) = a(5) = 1 because 1277 (resp. 88793) is the only prime with 4 (resp. 5) digits to start a prime octuplet.
Then there are a(6) = 3 six-digit primes, 113147, 284723 and 855713, which start a prime octuplet.

Links

Crossrefs

Cf. A065706 (initial members p of prime octuplets (p, ..., p+26)), A022011, A022012, A022013 (idem, specifically for each of the three possible patterns).
Cf. A350825, A350826, A350827: similar for quintuplets, sextuplets and septuplets.

Programs

  • PARI
    (D(v)=v[^1]-v[^-1])( [setsearch(A065706,10^n,1) | n<-[0..12]] ) \\ where A065706 is a vector of at least 3660 terms of that sequence.
Showing 1-4 of 4 results.

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