A006512 -id:A006512 - cp's OEIS frontend

A373128 Least k such that the k-th maximal antirun of squarefree numbers has length n. Position of first appearance of n in A373127.

1, 3, 10, 8, 19, 162, 1853, 2052, 1633, 26661, 46782, 3138650, 1080330

Offset: 1

Gus Wiseman, Jun 08 2024

An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of squarefree numbers begin:
   1
   2
   3   5
   6
   7  10
  11  13
  14
  15  17  19  21
  22
  23  26  29
  30
  31  33
  34
  35  37
The a(n)-th rows are:
    1
    3    5
   23   26   29
   15   17   19   21
   47   51   53   55   57
  483  485  487  489  491  493
For example, (23, 26, 29) is the first maximal antirun of 3 squarefree numbers, so a(3) = 10.

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

For composite instead of squarefree we have A073051.
Positions of first appearances in A373127.
The version for nonsquarefree runs is A373199, firsts of A053797.
For prime instead of squarefree we have A373401, firsts of A027833.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A120992, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[10000],SquareFreeQ[#]&],#1+1!=#2&]//Most;
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
Table[Position[t,k][[1,1]],{k,spnm[t]}]

A373401 Least k such that the k-th maximal antirun of prime numbers > 3 has length n. Position of first appearance of n in A027833. The sequence ends if no such antirun exists.

Original entry on oeis.org

1, 2, 4, 6, 10, 8, 69, 40, 24, 46, 41, 21, 140, 82, 131, 210, 50, 199, 35, 30, 248, 192, 277, 185, 458, 1053, 251, 325, 271, 645, 748, 815, 811, 1629, 987, 826, 1967, 423, 1456, 2946, 1109, 406, 1870, 1590, 3681, 2920, 3564, 6423, 1426, 5953, 8345, 12687, 6846

Offset: 1

Gus Wiseman, Jun 09 2024

nonn

The sorted version is A373402.

For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.

			The maximal antiruns of prime numbers > 3 begin:
    5
    7  11
   13  17
   19  23  29
   31  37  41
   43  47  53  59
   61  67  71
   73  79  83  89  97 101
  103 107
  109 113 127 131 137
  139 149
  151 157 163 167 173 179
The a(n)-th rows are:
     5
     7   11
    19   23   29
    43   47   53   59
   109  113  127  131  137
    73   79   83   89   97  101
  2269 2273 2281 2287 2293 2297 2309
  1093 1097 1103 1109 1117 1123 1129 1151
   463  467  479  487  491  499  503  509  521
For example, (19, 23, 29) is the first maximal antirun of length 3, so a(3) = 4.

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

For composite instead of prime we have A073051.
For runs instead of antiruns we have the triple (4,2,1), firsts of A251092.
For squarefree instead of prime we have A373128, firsts of A373127.
The sorted version is A373402.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
Cf. A001359, A005117, A027833, A006512, A053797, A373199, A373405.

t=Length/@Split[Select[Range[4,100000],PrimeQ],#1+2!=#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A107770 Index of greater of twin primes in the primes.

Original entry on oeis.org

3, 4, 6, 8, 11, 14, 18, 21, 27, 29, 34, 36, 42, 44, 46, 50, 53, 58, 61, 65, 70, 82, 84, 90, 99, 105, 110, 114, 117, 121, 141, 143, 145, 149, 153, 172, 174, 177, 179, 183, 191, 202, 207, 210, 213, 216, 226, 231, 235, 237, 254, 257, 263, 266, 269, 278, 287

Offset: 1

Roger L. Bagula, Jun 11 2005

Numbers k such that prime(k) - prime(k-1) = 2.

Numbers k such that A062301(k) is 1. - Vincenzo Librandi, Apr 04 2018

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

Cf. A062301.

[n: n in [1..450]| IsPrime(NthPrime(n)-2)]; // Vincenzo Librandi, Apr 04 2018

select(n->ithprime(n)-ithprime(n-1)=2, [$2..300]); # Muniru A Asiru, Apr 05 2018

a = Flatten[Table[If[Prime[j] - Prime[j - 1] == 2, j, {}], {j, 2, 200}]]
Flatten[Position[Partition[Prime[Range[400]],2,1],?(#[[2]]-#[[1]] == 2&),{1},Heads->False]]+1 (* _Harvey P. Dale, Jun 10 2014 *)

for(n=3, 1e3, if(isprime(prime(n)-2), print1(n, ", "))); \\ Altug Alkan, Apr 05 2018

a(n) = A029707(n) + 1. - Juri-Stepan Gerasimov, Dec 16 2009

a(n) = A000720(A006512(n)).

Incorrect comment removed by Charles R Greathouse IV, Mar 19 2010

More terms from Harvey P. Dale, Jun 10 2014

A166945 Records of first differences of A166944.

Original entry on oeis.org

2, 3, 7, 13, 43, 139, 313, 661, 1321, 2659, 5419, 10891, 22039, 44383, 88801, 177841, 355723, 713833, 1427749, 2860771, 5725453, 11461141, 22933441, 45895573, 91793059, 183616423, 367232911, 734482123, 1468965061, 2937930211, 5875882249, 11751795061, 23503590559, 47007181621, 94014363763

Offset: 1

Vladimir Shevelev, Oct 24 2009, Nov 05 2009

nonn

Conjecture. Each term of the sequence is the greater of a pair of twin primes (A006512).

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11(2008), Article 08.2.8. arXiv:0710.3217 [math.NT]
V. Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009.
V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010.

Cf. A166944, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

Reap[Print[old = r = 2]; Sow[old]; For[n = 2, n <= 10^6, n++, d = GCD[old, If[OddQ[n], n-2, n]]; If[d>r, r=d; Print[d]; Sow[d]]; old += d]][[2, 1]] (* Jean-François Alcover, Nov 03 2018, from PARI *)

print1(old=r=2); for(n=2,1e11, d=gcd(old,if(n%2,n-2,n)); if(d>r, r=d; print1(", "d)); old+=d) \\ Charles R Greathouse IV, Oct 13 2017

6 more terms from R. J. Mathar, Nov 19 2009; extension beginning with a(19) from Benoit Cloitre (private communication to Vladimir Shevelev)
a(25), a(26) from D. S. McNeil, Dec 13 2010
a(27)-a(30) from Charles R Greathouse IV, Oct 13 2017
a(31)-a(35) from Charles R Greathouse IV, Oct 17 2017

A194598 Union of A080359 and A164294.

Original entry on oeis.org

2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 157, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 269, 271, 283, 293, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 571, 577, 593, 599, 601

Offset: 1

Vladimir Shevelev, Aug 30 2011

nonn

Every greater of twin primes (A006512), beginning with 13, is in the sequence.
A very simple sieve for the generation of the terms is the following: Let p_n be the n-th prime. Consider consecutive intervals of the form (2p_n, 2p_{n+1}), n=1,2,... From every interval containing at least one prime we take the first one and remove it from the set of all primes. Then all remaining primes form the sequence. Let us demonstrate this sieve: For primes 2,3,5,7,11,... consider intervals (4,6), (6,10), (10,14), (14,22), (22,26), (26,34), ... . Removing from the set of all primes the first prime of each interval, i.e., 5,7,11,17,23,29,... ,we obtain 2,3,13,19,31, etc.
This sequence and A164368 are the mutually wrapping up sequences:
   a(1) <= A164368(1) <= a(2) <= A164368(2) <= ...
Following the steps to generate T(n,1) in A229608 provides an alternate method of generating this sequence. - Bob Selcoe, Oct 27 2015

Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009, 2011.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A080359, A164294, A164368, A193507, A194186, A212493, A212541, A229608.

If the first two terms are omitted we get A164333.

primePiMax = 200;
Join[{2, 3}, Select[Table[{(Prime[k-1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, primePiMax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ]&][[All, 2]]*2+1] (* Jean-François Alcover, Aug 18 2018 *)

First column of array A229608. - Bob Selcoe, Oct 27 2015

For n >= 3, a(n) = A164333(n-2). - Peter Munn, Aug 30 2017

A205649 Hamming distance between twin primes.

Original entry on oeis.org

2, 1, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 6, 1, 2, 4, 1, 1, 3, 2, 2, 4, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 7, 1, 1, 1, 1, 3, 2, 2, 1, 4, 3, 2, 2, 1, 1, 2, 4, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 1, 2, 1, 2, 1, 1, 5, 1, 7, 3, 1, 1, 1, 1, 3, 4, 5, 2, 1, 2

Offset: 1

Vladimir Shevelev, Jan 30 2012

Twin primes for which a(n)=1 are in A122565.

Conjecture: The sequence is unbounded.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A205510, A001097, A001359, A006512, A205302, A205511, A205533.

nn = 1000; ps = Prime[Range[nn]]; t = {}; Do[If[ps[[n]] + 2 == ps[[n + 1]], AppendTo[t, ps[[n]]]], {n, nn - 1}]; Table[b2 = IntegerDigits[t[[k]] + 2, 2];  b1 = IntegerDigits[t[[k]], 2, Length[b2]]; Total[Abs[b1 - b2]], {k, Length[t]}] (* T. D. Noe, Jan 30 2012 *)

A001359(n) == -1 (mod 2^a(n)).

A233547 a(n) = |{0 < k < n/2: phi(k)phi(n-k) - 1 and phi(k)phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 1, 3, 4, 3, 2, 3, 2, 3, 1, 1, 2, 1, 5, 2, 3, 1, 2, 1, 1, 3, 4, 5, 4, 3, 2, 3, 2, 5, 2, 5, 5, 3, 5, 3, 1, 5, 3, 7, 6, 3, 2, 4, 7, 5, 1, 4, 6, 6, 5, 2, 4, 6, 9, 9, 6, 8, 5, 8, 8, 6, 6, 9, 4, 8, 6, 8, 5, 7, 9, 7, 9, 5, 7, 3, 9, 5, 6, 7, 7, 10, 5, 12, 7, 5, 7, 5, 7, 5, 7, 8, 4, 7, 13

Offset: 1

Zhi-Wei Sun, Dec 12 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) For any n > 3, sigma(k)*phi(n-k) - 1 and sigma(k)*phi(n-k) + 1 are both prime for some 0 < k < n, where sigma(k) is the sum of all (positive) divisors of k.
(iii) For any n > 5 not equal to 35, there is a positive integer k < n such that phi(k)*phi(n-k) - 1 is a Sophie Germain prime.
Note that part (i) implies the twin prime conjecture. We have verified it for n up to 10^7.

			a(6) = 1 since phi(1)*phi(5) = 1*4 = 4 with 4 - 1 and 4 + 1 twin primes.
a(8) = 1 since phi(1)*phi(7) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(16) = 1 since phi(2)*phi(14) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(17) = 1 since phi(3)*phi(14) = 2*6 = 12 with 12 - 1 and 12 + 1 twin primes.
a(19) = 1 since phi(1)*phi(18) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(23) = 1 since phi(2)*phi(21) = 1*12 = 12 with 12 - 1 and 12 + 1 twin primes.
a(25) = 1 since phi(11)*phi(14) = 10*6 = 60 with 60 - 1 and 60 + 1 twin primes.
a(26) = 1 since phi(7)*phi(19) = 6*18 = 108 with 108 - 1 and 108 + 1 twin primes.
a(42) = 1 since phi(14)*phi(28) = 6*12 = 72 with 72 - 1 and 72 +1 twin primes.
a(52) = 1 since phi(14)*phi(38) = 6*18 = 108 with 108 - 1 and 108 + 1 twin primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A001359, A005384, A006512, A014574, A232270, A233544.

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
a[n_]:=Sum[If[TQ[EulerPhi[k]*EulerPhi[n-k]],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,100}]

A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).

Original entry on oeis.org

6, 8, 14, 14, 20, 20, 32, 32, 32, 44, 44, 44, 62, 62, 62, 62, 74, 74, 74, 104, 104, 104, 104, 104, 104, 110, 110, 140, 140, 140, 140, 140, 152, 152, 182, 182, 182, 182, 182, 182, 194, 194, 200, 200, 230, 230, 230, 230, 242, 242, 242, 272, 272, 272, 272, 272

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

This is a version of A242720 with the absolute minima of k in the definition. The sequence is nondecreasing. Hypothetically, every pair {a(n)-3, a(n)-1} is a pair of twin primes.
If there exist infinitely many n such that a(n) < A242719(n) < a(n)^2, then from the result in the Shevelev link, it follows that for such n the set of numbers {even k: lpf(k-1) > lpf(k-3) >= prime(n)} either attains the absolute minimum of a(n) only in the case when {a(n)-3, a(n)-1} are twin primes, or does not attain it at all. Therefore, if there is only a finite number of twin primes, we have a contradiction. Thus the above condition is sufficient for infinity of twin primes.
Note also that, if there is only a finite number of twin primes, then after the last pair of them, this sequence will coincide with A242720. Then, in order to avoid a contradiction (again according to the Shevelev link), we should accept that there exists a number N_0 such that, for every n >= N_0, the following inequality holds: max(A242719(n),A242720(n)) > (min(A242719(n),A242720(n)))^2. - Vladimir Shevelev, May 24 2014
It is easy to prove that min(A242719(n), A242720(n)) >= prime(n)^2+1, while we conjecture that max(A242719(n), A242720(n)) <= prime(n)^4. Thus this conjecture implies there are infinitely many twin primes. - Vladimir Shevelev, Jun 01 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..10001
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A001359, A006512, A242489, A242490, A242720.

lpf[k_] := FactorInteger[k][[1, 1]];
a[n_] := a[n] = For[k = If[n == 2, 2, a[n-1]], True, k = k+2, If[lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Nov 03 2018 *)

lpf(k) = factorint(k)[1,1];
vector(100, n, k=6; while(lpf(k-1)<=lpf(k-3) || lpf(k-3)Colin Barker, Jun 01 2014

A373821 Run-lengths of run-lengths of first differences of odd primes.

Original entry on oeis.org

1, 11, 1, 19, 1, 1, 1, 5, 1, 6, 1, 16, 1, 27, 1, 3, 1, 1, 1, 6, 1, 9, 1, 29, 1, 2, 1, 18, 1, 1, 1, 5, 1, 3, 1, 17, 1, 19, 1, 30, 1, 17, 1, 46, 1, 17, 1, 27, 1, 30, 1, 5, 1, 36, 1, 41, 1, 10, 1, 31, 1, 44, 1, 4, 1, 14, 1, 6, 1, 2, 1, 32, 1, 13, 1, 17, 1, 5

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Run-lengths of A333254.
The first term other than 1 at an odd positions is at a(101) = 2.
Also run-lengths (differing by 0) of run-lengths (differing by 0) of run-lengths (differing by 1) of composite numbers.

			The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences:
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with run-lengths:
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...
with run-lengths a(n).

Run-lengths of run-lengths of A046933(n) = A001223(n) - 1.
Run-lengths of A333254.
A000040 lists the primes.
A001223 gives differences of consecutive primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373406.
For composite runs: A005381, A008864, A054265, A176246, A251092, A373403.
Cf. A006560, A037201, A038664, A069010, A373824, A373825.

Length/@Split[Length /@ Split[Differences[Select[Range[3,1000],PrimeQ]]]//Most]//Most

A037073 Numbers k such that (6*k)^2 is the sum of a twin prime pair.

Original entry on oeis.org

1, 2, 7, 8, 12, 14, 15, 29, 34, 44, 51, 62, 68, 76, 79, 91, 99, 100, 107, 125, 142, 147, 156, 162, 163, 173, 190, 202, 212, 231, 245, 252, 253, 264, 295, 306, 317, 330, 331, 355, 366, 376, 377, 386, 397, 442, 448, 453, 462, 469, 481, 491, 498, 502, 516, 547

Offset: 1

G. L. Honaker, Jr.

nonn

			E.g. n=44 -> (6*44)^2 = 69696 = 34847 + 34849 (twin prime pair).

Amiram Eldar, Table of n, a(n) for n = 1..10000
C. K. Caldwell, Twin primes

Cf. A000290, A001359, A006512, A173165.

A152786 = 6*A037073. - Zak Seidov, Aug 20 2010

isa := n -> isprime(n) and isprime(n+2) and issqr(2*n+2):
select(isa, [$4..1000000]): map(n -> sqrt(2*n+2)/6, %); # Peter Luschny, Jan 05 2020

Select[Sqrt[Plus@@@Select[Partition[Prime[Range[4*10^5]],2,1],Differences[#]=={2} &]/36],IntegerQ] (* Jayanta Basu, May 26 2013 *)

is(n)=isprime(18*n^2-1)&&isprime(18*n^2+1) \\ M. F. Hasler, Oct 30 2023

a(n) = A173165(n)/3. - M. F. Hasler, Oct 30 2023

More terms from Jud McCranie, Dec 30 2000

cp's OEIS Frontend

A373128 Least k such that the k-th maximal antirun of squarefree numbers has length n. Position of first appearance of n in A373127.

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A373401 Least k such that the k-th maximal antirun of prime numbers > 3 has length n. Position of first appearance of n in A027833. The sequence ends if no such antirun exists.

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A107770 Index of greater of twin primes in the primes.

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A166945 Records of first differences of A166944.

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PARI

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A194598 Union of A080359 and A164294.

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A205649 Hamming distance between twin primes.

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A233547 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) - 1 and phi(k)*phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

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A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).

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A233547 a(n) = |{0 < k < n/2: phi(k)phi(n-k) - 1 and phi(k)phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.