A098414 -id:A098414 - cp's OEIS frontend

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

-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

A073648 Middle members of prime triples {p, p+2, p+6}.

Original entry on oeis.org

7, 13, 19, 43, 103, 109, 193, 229, 313, 349, 463, 643, 823, 859, 883, 1093, 1279, 1303, 1429, 1483, 1489, 1609, 1873, 1999, 2083, 2239, 2269, 2659, 2689, 3253, 3463, 3529, 3673, 3919, 4003, 4129, 4519, 4639, 4789, 4933, 4969, 5233, 5479, 5503, 5653, 6199

Offset: 1

Amarnath Murthy, Aug 09 2002

Zak Seidov, Table of n,a(n) for n=1,2380, a(n)<2*10^6

Cf. A073649, A073650.

Cf. A098412, A098414.

Transpose[Select[Partition[Prime[Range[850]],3,1],Differences[#]=={2,4}&]][[2]]  (* Harvey P. Dale, Feb 20 2011 *)

a(n) = A022004(n) + 2.

More terms from Benoit Cloitre, Aug 13 2002

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

Reinhard Zumkeller, Sep 07 2004

nonn

Union of A098412 and A098413;

a(n)=A007529(n)+6; either a(n)=A098414(n)+2 or a(n)=A098414(n)+4.

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

Cf. A098424, A098416, A098417.

Transpose[Select[Partition[Prime[Range[350]],3,1],#[[3]]- #[[1]] == 6&]][[3]] (* Harvey P. Dale, Mar 17 2015 *)

is(n)=isprime(n) && isprime(n-6) && (isprime(n-2) || isprime(n-4)) \\ Charles R Greathouse IV, Feb 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

Amarnath Murthy, Aug 09 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

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

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

Corrected and extended by Benoit Cloitre, Aug 13 2002

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.

A166007 Number of ones in the binary representation of the middle member q of the prime triple (p,q,r) with p

Original entry on oeis.org

3, 3, 3, 2, 3, 3, 4, 4, 4, 5, 5, 5, 3, 4, 5, 5, 4, 6, 5, 6, 6, 7, 5, 4, 7, 7, 6, 7, 6, 7, 4, 4, 9, 5, 6, 6, 6, 7, 7, 8, 6, 5, 5, 5, 9, 8, 6, 7, 8, 9, 4, 5, 6, 8, 7, 6, 6, 9, 4, 7, 7, 8, 7, 7, 6, 7, 7, 7, 7, 7, 9, 8, 3, 6, 6, 7, 7, 7, 7, 6, 7, 8, 6, 6, 5, 8

Offset: 1

Steven Lubars (lubars(AT)gmail.com), Oct 03 2009

			For n = 3, (p, q, r) = (11, 13, 17), q = 13
Decimal 13 = Binary 1101
a(3) = Number of ones in 1101 = 3

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A098414, A014499

DigitCount[#,2,1]&/@Transpose[Select[Partition[Prime[Range[1000]],3,1], Last[#]-First[#]==6&]][[2]] (* Harvey P. Dale, Dec 03 2014 *)

More terms from Harvey P. Dale, Dec 03 2014

Showing 1-10 of 11 results. Next

cp's OEIS Frontend

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

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

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A073648 Middle members of prime triples {p, p+2, p+6}.

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

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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).

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

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

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

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Haskell

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

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