A098415 -id:A098415 - cp's OEIS frontend

A098416 (A007529(n) + A098415(n)) / 4.

4, 5, 7, 8, 10, 20, 22, 35, 50, 52, 53, 55, 97, 98, 113, 115, 140, 155, 157, 175, 230, 232, 308, 322, 412, 413, 428, 430, 440, 442, 545, 547, 640, 650, 652, 713, 715, 725, 742, 743, 745, 805, 833, 848, 893, 935, 937, 938, 998, 1000, 1042, 1043, 1070, 1120, 1135

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

(A007529(n),2*a(n)+A098417(n), A098415(n)) is a prime triple (p,q,r) with p

Eric Weisstein's World of Mathematics, Prime Triplet

A098417 A098414(n) - (A007529(n) + A098415(n))/2.

Original entry on oeis.org

-1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, 1, -1

Offset: 1

Reinhard Zumkeller, Sep 07 2004

sign

(A007529(n),2*A098416(n)+a(n), A098415(n)) is a prime triple (p,q,r) with p

Eric Weisstein's World of Mathematics, Prime Triplet

A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 37, 41, 67, 97, 101, 103, 107, 191, 193, 223, 227, 277, 307, 311, 347, 457, 461, 613, 641, 821, 823, 853, 857, 877, 881, 1087, 1091, 1277, 1297, 1301, 1423, 1427, 1447, 1481, 1483, 1487, 1607, 1663, 1693, 1783, 1867, 1871, 1873, 1993, 1997

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Or, prime(m) such that prime(m+2) = prime(m)+6. - Zak Seidov, May 07 2012

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Cf. A031924, A023201, A098414, A098415, A098424, A098416, A098417.

Union of A022004 and A022005.

[NthPrime(n): n in [1..310] | (NthPrime(n)+6) eq NthPrime(n+2)]; // Bruno Berselli, May 07 2012

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(2*i+1, i=1..floor((N+5)/2))]):locs:= select(t -> Primes[t+2]-Primes[t]=6, [$1..nops(Primes)-2]):
Primes[locs]; # Robert Israel, Apr 30 2015

ptrsQ[n_]:=PrimeQ[n+6]&&(PrimeQ[n+2]||PrimeQ[n+4])
Select[Prime[Range[400]],ptrsQ]  (* Harvey P. Dale, Mar 08 2011 *)

p=2;q=3;forprime(r=5,1e4,if(r-p==6,print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, May 07 2012

a(n) = A098415(n) - 6. - Zak Seidov, Apr 30 2015

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

Reinhard Zumkeller, Sep 07 2004

nonn

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098413, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-4)]; // Vincenzo Librandi, Dec 26 2010

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

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 *)

A098413 Greatest members p of prime triples (p-6, p-2, p).

Original entry on oeis.org

13, 19, 43, 73, 103, 109, 199, 229, 283, 313, 463, 619, 829, 859, 883, 1093, 1303, 1429, 1453, 1489, 1669, 1699, 1789, 1873, 1879, 1999, 2089, 2143, 2383, 2689, 2713, 2803, 3169, 3259, 3463, 3469, 3853, 4159, 4519, 4789, 5233, 5419, 5443, 5653, 5659, 5743

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Subsequence of A046117; a(n) = A073649(n) + 2 = A022005(n) + 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098412, A098415.

[p: p in PrimesUpTo(6500)|IsPrime(p) and IsPrime(p-6) and IsPrime(p-2)]; // Vincenzo Librandi, Dec 26 2010

Transpose[Select[Partition[Prime[Range[800]],3,1],Differences[#] == {4,2}&]][[3]] (* Harvey P. Dale, Aug 21 2013 *)

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

Reinhard Zumkeller, Sep 07 2004

nonn

A098418(a(n)) > 0; complement of A098419 in A000040.

Union of A007529, A098414 and A098415.

Paul Shubhankar, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013
Paul Shubhankar, Legendre, Grimm, Balanced Prime, Prime triple, Polignac's conjecture, a problem and 17 tips with proof to solve problems on number theory, International Journal of Engineering and Technical Research (IJETR), Volume-1, Issue-10, December 2013.
Eric Weisstein's World of Mathematics, Prime Triplet

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

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

Reinhard Zumkeller, Sep 07 2004

nonn

0 <= a(n) <= 3;

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

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

Eric Weisstein's World of Mathematics, Prime Triplet

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

Reinhard Zumkeller, Sep 07 2004

nonn

Convention: a prime triple is <= n iff its smallest member is <= n;

a(n) <= A098428(n).

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A007529, A098414, A098415, A071538.

Cf. A010051, A000040.

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

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

Reinhard Zumkeller, Sep 07 2004

nonn

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

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

Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098419, A098421, A098422.

A055737 Number of prime triples < 10^n, where a prime triple means 3 successive primes of the form {p, p+2, p+4} or {p, p+4, p+6}.

Original entry on oeis.org

0, 8, 30, 112, 507, 2837, 17220, 111156, 759256, 5425573, 40174725, 305689269, 2379622234, 18887841658

Offset: 1

Robert G. Wilson v, Jun 09 2000

For this sequence all three members of the triple must be below the 10^n bound. - Sean A. Irvine, Apr 04 2022

J. Recreational Math., vol. 23, No. 2, 1991, p. 97.

Index entries for sequences related to numbers of primes in various ranges

Cf. A055738, A006880, A007529, A098415.

x=168; a=Table[ Prime[ n ], {n, 1, x} ] c=0; Do[ If[ a[[ n ]]+6==a[ [ n+2 ] ], c++ ], {n, 1, x-2} ]; c # the values of x to use are given by A006880

a(7)-a(9) from Jud McCranie, Oct 07 2000
a(10)-a(12) from Martin Raab, Oct 04 2006
a(13)-a(14) from Charles R Greathouse IV, Feb 09 2022

Showing 1-10 of 10 results.

cp's OEIS Frontend

A098416 (A007529(n) + A098415(n)) / 4.

Views

Author

Keywords

Comments

Links

A098417 A098414(n) - (A007529(n) + A098415(n))/2.

Views

Author

Keywords

Comments

Links

A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A098413 Greatest members p of prime triples (p-6, p-2, p).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

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

Views

Author

Keywords