A023201 -id:A023201 - cp's OEIS frontend

A121069 Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).

2, 4, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070

Offset: 1

Lekraj Beedassy, Aug 10 2006

If n > 2, then a(n) = product of n-1 consecutive distinct prime divisors. E.g. a(5)=210, the product of 4 consecutive and distinct prime divisors, 2,3,5,7. - Enoch Haga, Dec 08 2007
From Bill McEachen, Jul 10 2022: (Start)
Rather than have code merely generating the conjectured values, one can compare values of sequence terms at the same position n. Specifically, locate new maximums where (p,p+even) are both prime, where even=2,4,6,8,... and the datum set is taken with even=4. A new maximum implies a new jumping champion.
Doing this produces the terms 2,4,6,30,210,2310,30030,.... Looking at the plot of a(n) ratio for gap=2/gap=6, the value changes VERY slowly, and is 2.14 after 50 million terms (one can see the trend via Plot 2 of A001359 vs A023201 (3rd option seqA/seqB vs n). The ratio for gap=4/gap=2 ~ 1, implying they are equally frequent. (End)

R. P. Brent, The First Occurrence of Large Gaps Between Successive Primes
R. P. Brent, The distribution of small gaps between successive primes
R. P. Brent, The first occurrence of certain large prime gaps
C. K. Caldwell, The Prime Glossary, gaps between primes
C. K. Caldwell, The Prime Glossary, Jumping champion
S. Funkhouser, D. A. Goldston, D. Sengupta, and J. Sengupta, Prime Difference Champions, arXiv:1612.02938 [math.NT], 2016.
D. A. Goldston and A. H. Ledoan, Jumping champions and gaps between consecutive primes, Oct 15, 2009. [From _Jonathan Vos Post_, Oct 17 2009]
A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions
A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions
A. M. Odlyzko, M. Rubinstein, and M. Wolf, CHANCE News 10.02, 10. Jumping champions in the world of primes
A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions, Experiment. Math. 8(2): 107-118 (1999).
Tomás Oliveira e Silva, Gaps between consecutive primes
Ian Stewart, Jumping Champions, Scientific American, Vol. 283, No. 6 (2000), pp. 106-107; Wayback Machine link.
Eric Weisstein's World of Mathematics, Jumping Champion

Cf. A087103, A087104, A001223, A000230, A001632, A038664, A086977-A086980, A085237, A005250, A053686, A054587, A093737-A093753, A093972-A093984.

2,4,Table[Product[Prime[k],{k,1,n-1}],{n,3,30}]

print1("2, 4");t=2;forprime(p=3,97,print1(", ",t*=p)) \\ Charles R Greathouse IV, Jun 11 2011

Consists of 4 and the primorials (A002110).

a(1) = 2, a(2) = 4, a(3) = 6, a(n+1)/a(n) = Prime[n] for n>2.

Corrected and extended by Alexander Adamchuk, Aug 11 2006

Definition corrected and clarified by Jonathan Sondow, Aug 16 2011

A136207 Primes p such that p-6 or p+6 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337

Offset: 1

Carlos Alves, Dec 21 2007

nonn

Either or both of (p-6) and (p+6) is/are prime. - Harvey P. Dale, Jun 22 2019

Lei Zhou, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
Wikipedia, Sexy Primes

Cf. A023201, A046117, A140546 (complement).

isA136207 := proc(n)
    if isprime(n) then
        if isprime(n+6) or isprime(n-6) then
            true;
        else
            false;
        end if;
    else
        false ;
    end if;
end proc:
A136207 := proc(n)
    option remember;
    local a;
    if n = 1 then
        5 ;
    else
        a := nextprime(procname(n-1)) ;
        while true do
            if isA136207(a) then
                return a;
            else
                a := nextprime(a) ;
            end if;
        end do:
    end if;
end proc:
seq(A136207(n),n=1..80) ; # R. J. Mathar, Jun 10 2024

dd = 6; DistancePrimesQ1 = (PrimeQ[ # ] && PrimeQ[ # + dd]) &; DistancePrimesQ2 = (PrimeQ[ # ] && PrimeQ[ # - dd] && (# > dd)) &; DistancePrimesQQ = (DistancePrimesQ1[ # ] || DistancePrimesQ2[ # ]) &; DistancePrimes = Select[Range[ # ], DistancePrimesQQ] &; DistancePrimes[1000]
Alternative by Lei Zhou:
p = 3; Table[While[p = NextPrime[p]; ! (PrimeQ[p - 6] || PrimeQ[p + 6])]; p, {n, 1, 100}]
Select[Prime[Range[3,100]],AnyTrue[#+{6,-6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2019 *)

A252089 Primes p such that p + 26 is prime.

Original entry on oeis.org

3, 5, 11, 17, 41, 47, 53, 71, 83, 101, 113, 131, 137, 167, 173, 197, 251, 257, 281, 311, 347, 353, 383, 431, 461, 521, 587, 593, 617, 647, 683, 701, 743, 761, 797, 827, 857, 881, 911, 941, 971, 983, 1013, 1061, 1091, 1097, 1103, 1187, 1223, 1277, 1301, 1373

Offset: 1

Vincenzo Librandi, Dec 14 2014

			17 is in this sequence because 17+26 = 43 is prime.
431 is in this sequence because 431+26 = 457 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), this sequence (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), A156104 (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+26)];

Select[Prime[Range[200]], PrimeQ[# + 26] &]

A054903 Composite numbers n such that sigma(n)+6 = sigma(n+6), where sigma=A000203.

Original entry on oeis.org

104, 147, 596, 1415, 4850, 5337, 370047, 1630622, 35020303, 120221396, 3954451796, 742514284703

Offset: 1

Labos Elemer, May 23 2000

Complement of A023201 with respect to A015914.
Intersection of A015914 and A018252.
Below 1000000 there are only 7 such composite numbers, compared with more than 16000 primes.
a(13) > 10^13. - Giovanni Resta, Jul 11 2013

			n=104, sigma(104)+6 = 210+6 = 216 = sigma(104+6) = sigma(110).
a(4) = 1415 = 5*283, 1415+6 = 1421 = 7*7*29:
sigma(1415) = 1+5+283+1415 = 1704,
sigma(1421) = 1+7+29+49+203+1421 = 1710 = sigma(1415)+6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 104, p. 37, Ellipses, Paris 2008.

Cf. A015913-A015917, A023200-A023203, A046133, A001359, A054799.

forcomposite(n=9,1e7,if(sigma(n)+6==sigma(n+6),print1(n", "))) \\ Charles R Greathouse IV, Feb 14 2013

More terms from Jud McCranie, May 25 2000
New definition from Reinhard Zumkeller, Jan 27 2009
Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of R. J. Mathar.
a(12) from Giovanni Resta, Jul 11 2013

A093738 Number of pairs of consecutive prime (p,q) with q-p=6 and q < 10^n.

Original entry on oeis.org

0, 7, 44, 299, 1940, 13549, 99987, 768752, 6089791, 49392723, 408550278, 3435528229, 29289695650, 252672394234, 2201981901415, 19360330918473, 171550299264139, 1530609037414453

Offset: 1

Enoch Haga, Apr 15 2004

Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the former, whereas A080841 considers the latter. - N. J. A. Sloane, Mar 07 2021

			a(2) = 7 because there are 7 prime gaps of 6 below 10^2.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A080841, A093737, A093739.

A104229 Primes equal to the product of two successive sexy primes plus 6.

Original entry on oeis.org

61, 97, 193, 397, 673, 1153, 1597, 1933, 4093, 7393, 12097, 37633, 64513, 70753, 96097, 122497, 126733, 136897, 190093, 211597, 256033, 313597, 329473, 348097, 430333, 541693, 781453, 891133, 988033, 1267873, 1416097, 1674433, 2102497

Offset: 1

Giovanni Teofilatto, Apr 02 2005

Primes of the form 6 + A111192(i). - R. J. Mathar, Nov 26 2008

All numbers in this sequence are of the form 12n + 1. Also, as one would expect from a random distribution of sexy prime pairs, with the exception of 61, in decimal two thirds of these numbers end in 3, and the other third end in 7. - Daniel Mondot, Apr 29 2024

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A023201, A046117, A104227, A104228, A111192.

Extended by R. J. Mathar, Nov 26 2008

A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

Original entry on oeis.org

55, 91, 187, 247, 275, 391, 605, 637, 667, 1147, 1183, 1375, 1591, 1927, 2057, 2491, 3025, 3127, 3179, 3211, 4087, 4459, 4693, 4891, 5767, 6647, 6655, 6875, 7387, 8281, 8993, 9991, 10807, 11227, 12091, 15125, 15341, 15379, 17947, 19343, 22627, 23707

Offset: 1

Reinhard Zumkeller, Jul 30 2008

nonn

Subsequence of A007774.
A111192 is a subsequence.
Subsequence of A195118. - Reinhard Zumkeller, Sep 13 2011

			a(1) = 55 = 5 * 11 = A023201(1) * A046117(1).
a(2) = 91 = 7 * 13 = A023201(2) * A046117(2).
a(3) = 187 = 11 * 17 = A023201(3) * A046117(3).
a(4) = 247 = 13 * 19 = A023201(4) * A046117(4).
a(5) = 275 = 5^2 * 11 = A023201(1)^2 * A046117(1).
a(6) = 391 = 17 * 23 = A023201(5) * A046117(5).
a(7) = 605 = 5 * 11^2 = A023201(1) * A046117(1)^2.
a(8) = 637 = 7^2 * 13 = A023201(2)^2 * A046117(2).
a(9) = 667 = 23 * 29 = A023201(6) * A046117(6).
a(10) = 1147 = 31 * 37 = A023201(7) * A046117(7).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
Index entries for primes, gaps between.

Cf. A001221, A006530, A007774, A020639, A023201, A027748, A046117, A111192, A143201, A195118.

a143205 n = a143205_list !! (n-1)
a143205_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 6 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

okQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[1]]},Length[fi]==2 && Last[fi]-First[fi]==6]; Select[Range[25000],okQ]  (* Harvey P. Dale, Apr 18 2011 *)

A143201(a(n)) = 7.
A020639(a(n)) in A023201 and A006530(a(n)) in A046117.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023201(n)+2)^2-9) = 0.058842810164... . - Amiram Eldar, Oct 26 2024

A156274 List of prime pairs of the form (p, p+6).

Original entry on oeis.org

5, 11, 7, 13, 11, 17, 13, 19, 17, 23, 23, 29, 31, 37, 37, 43, 41, 47, 47, 53, 53, 59, 61, 67, 67, 73, 73, 79, 83, 89, 97, 103, 101, 107, 103, 109, 107, 113, 131, 137, 151, 157, 157, 163, 167, 173, 173, 179, 191, 197, 193, 199, 223, 229, 227, 233, 233, 239, 251, 257

Offset: 1

Vincenzo Librandi, Feb 07 2009

A023201 and A046117 interleaved. [From R. J. Mathar, Feb 19 2009]

			For p=5, p+6=11, (5,11); p=7, p+6=13, (7,13); p=11, p+6=17, (11,17)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Flatten[Select[{#, # + 6}&/@Prime[Range[100]], PrimeQ[Last[#]]&]] (* Vincenzo Librandi, Apr 06 2013 *)

A227346 Distance between consecutive pairs of primes differing by 6 (p, p+6).

Original entry on oeis.org

2, 4, 2, 4, 6, 8, 6, 4, 6, 6, 8, 6, 6, 10, 14, 4, 2, 4, 24, 20, 6, 10, 6, 18, 2, 30, 4, 6, 18, 6, 6, 8, 6, 30, 4, 20, 16, 6, 14, 6, 10, 50, 10, 14, 4, 42, 38, 16, 6, 8, 16, 6, 8, 6, 6, 28, 6, 6, 24, 50, 6, 18, 70, 2, 30, 4, 20, 4, 60, 6, 24, 6, 14, 22, 20, 30

Offset: 1

Luca Pezzullo, Jul 08 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A023201 (n and n+6 are primes).

Cf. A053320 (differences for Cousin primes).

with(numtheory): pre:=0: for n from 1 to 3000 do if isprime(n) and isprime(n+6) then if pre<>0 then printf("%d, ", n-pre) fi: pre:=n fi od: # adapted from original program by C. Ronaldo for A053320

Differences[Select[Prime[Range[200]], PrimeQ[# + 6] &]] (* T. D. Noe, Jul 09 2013 *)

a(n) = A023201(n+1) - A023201(n). - Zak Seidov, Sep 20 2013

A227898 Number of primes p < n with p + 6 and n + (n - p)^2 both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) For any integer n > 5, there is a prime p with p + 6 and n*(n - p) - 1 both prime.

			a(6) = 1 since 5, 5 + 6 = 11 and 6 + (6 - 5)^2 = 7 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A023201, A185636, A204065, A227899, A230261.

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[n+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A121069 Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A136207 Primes p such that p-6 or p+6 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A252089 Primes p such that p + 26 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A054903 Composite numbers n such that sigma(n)+6 = sigma(n+6), where sigma=A000203.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Extensions

A093738 Number of pairs of consecutive prime (p,q) with q-p=6 and q < 10^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

UBASIC

Extensions

A104229 Primes equal to the product of two successive sexy primes plus 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A156274 List of prime pairs of the form (p, p+6).

Views

Author

Keywords