A029707 -id:A029707 - cp's OEIS frontend

A155752 Where 2's occur in the prime differences A001223.

1, 2, 4, 6, 9, 12, 16, 19, 25, 27, 32, 34, 40, 42, 44, 48, 51, 56, 59, 63, 68, 80, 82, 88, 97, 103, 108, 112, 115, 119, 139, 141, 143, 147, 151, 170, 172, 175, 177, 181, 189, 200, 205, 208, 211, 214, 224, 229, 233, 235, 252, 255, 261, 264, 267, 276, 285, 287, 293, 295, 301

Offset: 1

Paul Curtz, Jan 26 2009

nonn

Starts with the same numbers as A053096.

Flatten[Position[Differences[Prime[Range[400]]],2]]-1 (* Harvey P. Dale, Jun 21 2017 *)

A001223(1+a(n)) = 2.

a(n) = A029707(n)-1. - R. J. Mathar, Feb 23 2009

Edited and extended by R. J. Mathar, Feb 23 2009

A242490 Smallest even number k such that lpf(k-3) = prime(n) while lpf(k-1) > lpf(k-3), where lpf=least prime factor (A020639).

Original entry on oeis.org

6, 8, 80, 14, 224, 20, 440, 854, 32, 1460, 1742, 44, 2282, 3434, 4190, 62, 5432, 4760, 74, 12194, 8930, 8054, 12374, 13292, 104, 15350, 110, 14282, 31982, 17402, 18212, 140, 24050, 152, 25220, 29990, 28202, 32234, 33392, 182, 43262, 194, 44972, 200, 47564

Offset: 2

Vladimir Shevelev, May 16 2014

nonn

Note that the "small terms" {6,8,14,20,32,44,...} correspond to a(n) for which {a(n)-3, a(n)-1} is a twin pair such that the corresponding positions form sequence A029707.

If we change the definition to consider k for which {k-3, k-1} is not a twin pair, we obtain a closely related sequence 12,38,80,212,224,530,440,854,1250,1460,1742,... which shows a "model behavior" of A242490, if there are only a finite number of twin primes. - Vladimir Shevelev, May 19 2014

			Let n=2, prime(2)=3. Then lpf(6-3)=3, but lpf(6-1)=5>3. Since k=6 is the smallest such k, a(2)=6.

Peter J. C. Moses, Table of n, a(n) for n = 2..1001

Cf. A001359, A006512, A242489.

a(n)=my(p=prime(n),k=p+3); while(factor(k-3)[1,1]Charles R Greathouse IV, May 30 2014

Correction and more terms from Peter J. C. Moses, May 19 2014

A373825 Position of first appearance of n in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

Original entry on oeis.org

1, 2, 13, 11, 105, 57, 33, 69, 59, 29, 227, 129, 211, 341, 75, 321, 51, 45, 407, 313, 459, 301, 767, 1829, 413, 537, 447, 1113, 1301, 1411, 1405, 2865, 1709, 1429, 3471, 709, 2543, 5231, 1923, 679, 3301, 2791, 6555, 5181, 6345, 11475, 2491, 10633

Offset: 1

Gus Wiseman, Jun 21 2024

nonn

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 positions of first appearances a(n).

Firsts of A373819 (run-lengths of A251092).
For antiruns we have A373827 (sorted A373826), firsts of A373820, run-lengths of A027833 (partial sums A029707, firsts A373401, sorted A373402).
The sorted version is A373824.
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.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A176246, A373403, A373404.
Cf. A006560, A006562, A037201, A038664, A069010, A122535.

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]}]

A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.

Original entry on oeis.org

4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Labos Elemer

nonn

Numbers k such that prime(k+1) - prime(k) does not divide prime(k+1) + prime(k). These are the numbers k for which prime(k+1) - prime(k) > 2. - Thomas Ordowski, Mar 31 2022

If we prepend 1, the first differences are A251092 (see also A175632). The complement is A029707. - Gus Wiseman, Dec 03 2024

			prime(4) = 7, 6!+1 = 721 gives residue 1 when divided by prime(4)+2 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The first differences are A251092 except first term, run-lengths A373819.
The complement is A029707.
Runs of terms differing by one have lengths A027833, min A107770, max A155752.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A038664 finds the first prime gap of difference 2n.
A046933 counts composite numbers between primes.
A071148 gives partial sums of odd primes.
For prime runs: A001359, A005381, A006512, A025584, A067774, A373403.
Cf. A006560, A006562, A037201, A122535, A175632, A176246, A373820.

pnmQ[n_]:=Module[{p=Prime[n]},Mod[(p-1)!+1,p+2]==1]; Select[Range[ 100],pnmQ] (* Harvey P. Dale, Jun 24 2017 *)

isok(n) = (((prime(n)-1)! + 1) % (prime(n)+2)) == 1; \\ Michel Marcus, Dec 31 2013

Definition edited by Thomas Ordowski, Mar 31 2022

A373819 Run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 10, 2, 4, 1, 7, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 18, 3, 2, 1, 2, 1, 17, 2, 1, 2, 2, 1, 6, 1, 9, 1, 3, 1, 1, 1, 1, 1, 1, 1, 8, 1, 3, 1, 2, 2, 15, 1, 1, 1, 4, 1, 1, 1, 1, 1, 7, 1

Offset: 1

Gus Wiseman, Jun 20 2024

nonn

Run-lengths of A251092.

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with runs:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
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, ...
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 a(n).

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Run-lengths of A251092.
For antiruns we have A373820, run-lengths of A027833 (if we prepend 1).
Positions of first appearances are A373825, sorted A373824.
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.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A373403, A373404.
Cf. A005117, A006560, A006562, A029707, A037201, A038664, A076259, A176246.

Length/@Split[Length/@Split[Select[Range[3,1000], PrimeQ],#1+2==#2&]//Most]//Most

A109374 Irregular table read by rows: Row n is the terms of the continued fraction for prime(n+1)/prime(n).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 5, 2, 1, 3, 4, 1, 8, 2, 1, 4, 1, 3, 1, 3, 1, 5, 1, 14, 2, 1, 5, 6, 1, 9, 4, 1, 20, 2, 1, 10, 1, 3, 1, 7, 1, 5, 1, 8, 1, 5, 1, 29, 2, 1, 10, 6, 1, 16, 1, 3, 1, 35, 2, 1, 12, 6, 1, 19, 1, 3, 1, 13, 1, 5, 1, 11, 8, 1, 24, 4, 1, 50, 2, 1, 25, 1, 3, 1, 53, 2, 1, 27, 4, 1

Offset: 1

Leroy Quet, Aug 24 2005

Sequence A071866 gives the number of terms in the n-th continued fraction.

If n is in A029707, row n is [1, (prime(n)-1)/2, 2]. - Robert Israel, May 29 2018

			Prime(4)/prime(3) = 7/5 = 1+ 1/(2+1/2), so the terms associated with the 3rd continued fraction are 1, 2, 2.

Robert Israel, Table of n, a(n) for n = 1..10003 (rows 1 to 2533, flattened)

Cf. A029707, A071866, A110021, A112323, A112324, A112768.

seq(op(convert(ithprime(n+1)/ithprime(n), confrac)),n=1..100); # Robert Israel, May 29 2018

Flatten[Table[ContinuedFraction[Prime[n + 1]/Prime[n]], {n, 30}]] (* Ray Chandler, Aug 25 2005 *)

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005
Edited by Charles R Greathouse IV, Apr 23 2010
Definition corrected by Leroy Quet, May 10 2010

A320703 Indices of primes followed by a gap (distance to next larger prime) of 10.

Original entry on oeis.org

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 443, 474, 491, 497, 500, 502, 518, 525, 531, 535, 575

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031928.

Vincenzo Librandi, Table of n, a(n) for n = 1..8870
Index entries for primes, gaps between.

Equals A000720 o A031928.
Row 5 of A174349.
Indices of 10's in A001223.
Subsequence of A107730: prime(n+1) ends in same digit as prime(n).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1000] | NthPrime(n+1) - NthPrime(n) eq 10]; // Vincenzo Librandi, Mar 21 2019

Select[Range[1000], Prime[#] + 10 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)
Position[Partition[Prime[Range[600]],2,1],?(#[[2]]-#[[1]]==10&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Mar 08 2020 *)

A320703_vec(N=100,g=10,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031928(n)).

A320703 = { i > 0 | prime(i+1) = prime(i) + 10 }.

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

Original entry on oeis.org

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[[#]]]&]

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

A270192 Numbers n for which (prime(n+1)-prime(n)) mod 3 = 2.

Original entry on oeis.org

2, 3, 5, 7, 10, 13, 17, 20, 24, 26, 28, 30, 33, 35, 41, 43, 45, 49, 52, 57, 60, 62, 64, 66, 69, 72, 77, 79, 81, 83, 87, 89, 92, 94, 98, 104, 109, 113, 116, 120, 124, 126, 128, 132, 135, 137, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 162, 166, 171, 173, 176, 178, 182, 186, 190, 192, 196, 201, 206, 209, 212, 215, 220, 223, 225

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

			2 is present as prime(3) - prime(2) = 5 - 3 = 2 = 2 modulo 3.
24 is present as prime(24) = 89, prime(25) = 97 and 97-89 = 8 = 2 modulo 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequence of A270189.
Positions of 2's in A137264.
Cf. A000040, A001223, A270190, A270191.
Differs from its subsequence A029707 for the first time at n=9.

Select[Range@ 225, Mod[Prime[# + 1] - Prime@ #, 3] == 2 &] (* Michael De Vlieger, Mar 17 2016 *)
Position[Differences[Prime[Range[300]]],?(Mod[#,3]==2&)]//Flatten (* _Harvey P. Dale, Jul 22 2016 *)

isok(n) = ((prime(n+1) - prime(n)) % 3) == 2; \\ Michel Marcus, Mar 17 2016

cp's OEIS Frontend

A155752 Where 2's occur in the prime differences A001223.

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A242490 Smallest even number k such that lpf(k-3) = prime(n) while lpf(k-1) > lpf(k-3), where lpf=least prime factor (A020639).

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A373825 Position of first appearance of n in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

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A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.

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A373819 Run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

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A109374 Irregular table read by rows: Row n is the terms of the continued fraction for prime(n+1)/prime(n).

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A320703 Indices of primes followed by a gap (distance to next larger prime) of 10.

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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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