A038664 -id:A038664 - cp's OEIS frontend

A373824 Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

1, 2, 11, 13, 29, 33, 45, 51, 57, 59, 69, 75, 105, 129, 211, 227, 301, 313, 321, 341, 407, 413, 447, 459, 537, 679, 709, 767, 1113, 1301, 1405, 1411, 1429, 1439, 1709, 1829, 1923, 2491, 2543, 2791, 2865, 3301, 3471, 3641, 4199, 4611, 5181, 5231, 6345, 6555

Offset: 1

Gus Wiseman, Jun 21 2024

nonn

Sorted positions of first appearances in A373819.

			The runs of odd primes differing by 2 begin:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...
which have runs beginning:
  3
  2 2
  1
  2
  1
  2
  1 1
  2
  1
  2
  1 1 1 1
  2 2
  1 1 1
with lengths:
1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3,...
with sorted positions of first appearances a(n).

Sorted firsts of A373819 (run-lengths of A251092).
The unsorted version is A373825.
For antiruns we have A373826, unsorted A373827.
A000040 lists the primes.
A001223 gives differences of consecutive primes (firsts A073051), run-lengths A333254 (firsts A335406), run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths, run-lengths of A027833.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A373403, A373404.
Cf. A006560, A006562, A029707, A037201, A038664, A069010, A122535, A176246, A373401, A373402.

t=Length/@Split[Length/@Split[Select[Range[3,10000],PrimeQ],#1+2==#2&]];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

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

Original entry on oeis.org

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

A179067 Orders of consecutive clusters of twin primes.

Original entry on oeis.org

1, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Franz Vrabec, Jun 27 2010

For k>=1, 2k+4 consecutive primes P1, P2, ..., P2k+4 defining a cluster of twin primes of order k iff P2-P1 <> 2, P4-P3 = P6-P5 = ... = P2k+2 - P2k+1 = 2, P2k+4 - P2k+3 <> 2.

Also the lengths of maximal runs of terms differing by 2 in A029707 (leading index of twin primes), complement A049579. - Gus Wiseman, Dec 05 2024

			The twin prime cluster ((101,103),(107,109)) of order k=2 stems from the 2k+4 = 8 consecutive primes (89, 97, 101, 103, 107, 109, 113, 127) because 97-89 <> 2, 103-101 = 109-107 = 2, 127-113 <> 2.
From _Gus Wiseman_, Dec 05 2024: (Start)
The leading indices of twin primes are:
  2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, ...
with maximal runs of terms differing by 2:
  {2}, {3,5,7}, {10}, {13}, {17}, {20}, {26,28}, {33,35}, {41,43,45}, {49}, {52}, ...
with lengths a(n).
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A077800.
Cf. A001359, A111950, A087641.
Cf. A035789, A035790, A035791, A035792, A035793, A035794, A035795.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A006512 gives the greater of twin primes.
A029707 gives the leading index of twin primes, complement A049579.
A038664 finds the first prime gap of length 2n.
A046933 counts composite numbers between primes.
A000720, A006560, A006562, A014574, A037201, A107770, A122535, A155752, A175632, A251092.

R:= 1: count:= 1: m:= 0:
q:= 5: state:= 1:
while count < 100 do
 p:= nextprime(q);
 if state = 1 then
    if p-q = 2 then state:= 2; m:= m+1;
    else
      if m > 0 then R:= R,m; count:= count+1; fi;
      m:= 0
    fi
 else state:= 1;
 fi;
 q:= p
od:
R; # Robert Israel, Feb 07 2023

Length/@Split[Select[Range[2,100],Prime[#+1]-Prime[#]==2&],#2==#1+2&] (* Gus Wiseman, Dec 05 2024 *)

a(n)={my(o,P,L=vector(3));n++;forprime(p=o=3,,L=concat(L[2..3],-o+o=p);L[3]==2||next;L[1]==2&&(P=concat(P,p))&&next;n--||return(#P);P=[p])} \\ M. F. Hasler, May 04 2015

More terms from M. F. Hasler, May 04 2015

A086977 Increasing peaks in the prime gap sequence A000230.

Original entry on oeis.org

199, 1831, 5591, 30593, 81463, 82073, 162143, 173359, 404597, 542603, 544279, 1100977, 1444309, 2238823, 5845193, 6752623, 6958667, 11981443, 13626257, 49269581, 83751121, 147684137, 166726367, 378043979, 895858039, 1872851947

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the smaller of the two consecutive primes having a late occurring prime gap g = p_k+1 - p_k. All even gaps smaller than g occur at a smaller prime. Also, the next even gap g+2 also occurs earlier.

			1831 is in this list because the next prime is 1847, giving a prime gap of 16. All even gaps less than 16 occur before this (for smaller primes) and the next even gap, 18, also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A000230, A001223, A001632, A038664, A086978, A086979, A086980, A002386.

lst={};b=max=2;Do[a=2;While[NextPrime@a-a!=2n,a=NextPrime@a];If[a=max,AppendTo[lst,b]];b=a;If[b>max,max=b],{n,40}];lst (* Giorgos Kalogeropoulos, Aug 18 2021 *)

A086980 Late occurring prime gaps in the prime gap sequence A001223.

Original entry on oeis.org

12, 16, 32, 38, 46, 56, 66, 70, 74, 80, 88, 94, 102, 108, 116, 124, 134, 144, 150, 158, 166, 186, 194, 200, 228, 256, 264, 278, 294, 298, 316, 328, 334, 362, 370, 388, 422, 436, 442, 452, 466, 472, 482, 488, 510, 520, 536, 568, 576, 580, 590, 608, 628, 632

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the gap g = p_k+1 - p_k between consecutive primes with all even gaps smaller than g occurring at a smaller prime and the next even gap g+2 also occurring earlier.

			16 is in this list because the first time a prime gap of 16 occurs is between consecutive primes 1831 and 1847. All even prime gaps less than 16 occur for a smaller prime. The next even prime gap of 18 also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A000230, A001223, A001632, A038664, A086977, A086978, A086979, A002386.

A261525 a(n) = smallest m such that A031131(m) = 2*n.

Original entry on oeis.org

2, 3, 9, 8, 15, 23, 47, 29, 66, 114, 46, 220, 188, 258, 640, 375, 480, 589, 216, 326, 367, 1006, 738, 1183, 1985, 1847, 1662, 2224, 3731, 3861, 3561, 2699, 3792, 4521, 2225, 12541, 3384, 12761, 3385, 4058, 10228, 15747, 15927, 14357, 18280, 19025, 14123

Offset: 2

Reinhard Zumkeller, Aug 23 2015

nonn

A031131(a(n)) = 2*n and A031131(m) != 2*n for m < a(n);
A046931(n) = A000040(a(n)+1);
a(n)-th and (a(n)+2)-nd primes are the first pair that differ by 2*n;
conjecture: sequence is defined for all n > 1.

Reinhard Zumkeller, Table of n, a(n) for n = 2..100

Cf. A031131, A038664, A046931.

a261525 = (+ 1) . fromJust . (`elemIndex` a031131_list) . (* 2)

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

Original entry on oeis.org

2, 6, 15, 79, 68, 121, 162, 445, 416, 971, 836, 987, 2888, 1891, 1650, 5637, 5518, 4834, 9237, 8152, 10045, 21550, 20248, 20179, 29914, 36070, 24237, 53355, 52873, 34206, 103134, 90190, 63755, 147861, 98103, 117467, 209102, 206423, 124954, 237847, 369223

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

			We need the first 15 prime gaps (1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6) before we reach the 3rd appearance of 6, so a(6) = 15.

The first appearances are at A038664, seconds A356221.
Diagonal of A356222.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts divisors with gapless prime indices, complement A356225.
A356226 = gapless interval lengths of prime indices, run-lengths A287170.
Cf. A000005, A028334, A029709, A060681, A119313, A137921, A193829, A274121.

nn=1000;
gaps=Differences[Array[Prime,nn]];
Table[Position[gaps,2*n][[n,1]],{n,Select[Range[nn],Length[Position[gaps,2*#]]>=#&]}]

A373826 Sorted positions of first appearances in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

Original entry on oeis.org

1, 4, 38, 6781, 23238, 26100

Offset: 1

Gus Wiseman, Jun 22 2024

Sorted positions of first appearances in A373820 (run-lengths of A027833 with 1 prepended).

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with antiruns (differing by > 2):
(3), (5), (7,11), (13,17), (19,23,29), (31,37,41), (43,47,53,59), ...
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, ...
which have runs:
(1,1), (2,2), (3,3), (4), (3), (6), (2), (5), (2), (6), (2,2), (4), ...
with lengths:
2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
with sorted positions of first appearances a(n).

Sorted positions of first appearances in A373820, cf. A027833.
For runs we have A373824 (unsorted A373825), sorted firsts of A373819.
The unsorted version is A373827.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A001359, A006512, A025584, A037201, A038664, A054265, A067774, A176246, A251092, A373404, A373405, A373406.

t=Length/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ],#1+2!=#2&]];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A373827 Position of first appearance of n in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

Original entry on oeis.org

4, 1, 38, 6781, 26100, 23238

Offset: 1

Gus Wiseman, Jun 22 2024

Positions of first appearances in A373820 (run-lengths of A027833 with 1 prepended).

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with antiruns (differing by > 2):
(3), (5), (7,11), (13,17), (19,23,29), (31,37,41), (43,47,53,59), ...
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, ...
which have runs:
(1,1), (2,2), (3,3), (4), (3), (6), (2), (5), (2), (6), (2,2), (4), ...
with lengths:
2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
with positions of first appearances a(n).

Positions of first appearances in A373820.
For runs instead of antiruns we have A373825, sorted A373824.
The sorted version is A373826.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A001359, A006512, A025584, A027833, A037201, A038664, A054265, A067774, A176246, A251092, A373404, A373405, A373406.

t=Length/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ],#1+2!=#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A086978 Increasing peaks in the prime gap sequence A001632.

Original entry on oeis.org

211, 1847, 5623, 30631, 81509, 82129, 162209, 173429, 404671, 542683, 544367, 1101071, 1444411, 2238931, 5845309, 6752747, 6958801, 11981587, 13626407, 49269739, 83751287, 147684323, 166726561, 378044179, 895858267, 1872852203

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the larger of the two consecutive primes having a late occurring prime gap g = p_k+1 - p_k. All even gaps smaller than g occur at a smaller prime. Also, the next even gap g+2 also occurs earlier.

			1847 is in this list because the previous prime is 1831, giving a prime gap of 16. All even gaps less than 16 occur before this (for smaller primes) and the next even gap, 18, also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A000230, A001223, A001632, A038664, A086977, A086979, A086980, A002386.

cp's OEIS Frontend

A373824 Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

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Mathematica

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

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A179067 Orders of consecutive clusters of twin primes.

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Maple

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A086977 Increasing peaks in the prime gap sequence A000230.

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Mathematica

A086980 Late occurring prime gaps in the prime gap sequence A001223.

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A261525 a(n) = smallest m such that A031131(m) = 2*n.

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Haskell

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

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A373826 Sorted positions of first appearances in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

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