A091592 -id:A091592 - cp's OEIS frontend

A091591 Number of pairs of twin primes between n^2 and (n+1)^2.

1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 0, 3, 2, 0, 1, 3, 2, 0, 3, 2, 1, 3, 0, 3, 2, 1, 3, 2, 4, 2, 2, 3, 0, 2, 2, 4, 0, 2, 1, 1, 5, 4, 4, 1, 2, 3, 4, 3, 5, 2, 2, 3, 2, 4, 1, 2, 2, 3, 4, 3, 0, 3, 3, 2, 4, 5, 2, 2, 3, 4, 1, 2, 3, 2, 3, 3, 1, 5, 1, 3, 4, 4, 2, 5, 3, 4, 1, 3, 5, 1, 2

Offset: 3

Hugo Pfoertner, Jan 22 2004

a(1) and a(2) are omitted because they are dependent on the treatment of the twin pair (3,5). It is conjectured that a(n)>0 for all n>122. Proving this would also prove the twin prime conjecture.

Proving a(n)>0 for n>122 would also prove Legendre's conjecture that there is a prime between n^2 and (n+1)^2. - T. D. Noe, Feb 28 2007

			a(3)=1 because the interval [3^2,4^2] contains one pair of twins (11,13).
a(9)=0 because the interval [9^2,10^2] is one of the few known intervals (given in A091592) not containing twin primes.

T. D. Noe, Table of n, a(n) for n=3..10000
Eric Weisstein's World of Mathematics, Twin Prime Conjecture.

Cf. A000290, A001359, A006512, A091592.

Cf. A014085 (number of primes between n^2 and (n+1)^2)

a[n_] := (k = 0; For[p = NextPrime[n^2], p <= NextPrime[(n + 1)^2, -2], q = NextPrime[p]; If[q - p == 2, k++; p = NextPrime[q], p = q]]; k); Table[a[n], {n, 3, 107}] (* Jean-François Alcover, Jun 13 2012 *)
With[{tps=Select[Partition[Prime[Range[2000]],2,1],Last[#]-First[#] == 2&]},Table[ Count[tps,?(#[[1]]>n^2&&#[[2]]<(n+1)^2&)],{n,3,110}]] (* _Harvey P. Dale, Feb 19 2013 *)

A138205 Least number k such that n^2+k and n^2+k+2 are the smallest twin primes between squares n^2 and (n+1)^2, or 0 if there is no such k.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 10, 7, 0, 1, 16, 5, 10, 1, 2, 13, 22, 23, 0, 19, 20, 37, 40, 23, 16, 0, 0, 25, 16, 0, 58, 7, 2, 0, 4, 5, 58, 7, 0, 7, 16, 23, 22, 13, 2, 13, 28, 5, 0, 49, 56, 7, 0, 53, 94, 31, 2, 7, 46, 71, 46, 7, 32, 31, 4, 65, 28, 13, 26, 31, 58, 47, 88, 1, 14, 73, 0, 5, 28, 49, 8, 37

Offset: 1

T. D. Noe, Mar 06 2008

nonn

Checking up to n=10^6, it appears that a(n)=0 for only n=1 and the n in A091592.

			a(7)=10 because (59,61) is the smallest twin-prime pair between 49 and 64.

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

Cf. A091591 (number of twin prime pairs between n^2 and (n+1)^2).

Table[n2=n^2; k=1; While[k<2n+1 && !(PrimeQ[n2+k] && PrimeQ[n2+k+2]), k++ ]; If[k==2n+1,0,k], {n,10^4}]

A143509 First of a pair of twin primes of the form n^2+x with x<2*n, or 0 if no such prime exist.

Original entry on oeis.org

2, 5, 11, 17, 29, 41, 59, 71, 0, 107, 137, 149, 179, 197, 227, 269, 311, 347, 0, 419, 461, 521, 569, 599, 641, 0, 0, 809, 857, 0, 1019, 1031, 1091, 0, 1277, 1301, 1427, 1451, 0, 1607, 1697, 1787, 1871, 1949, 2027, 2129, 2237, 2309, 0, 2549, 2657, 2711, 0, 2969

Offset: 1

Pierre CAMI, Aug 21 2008

nonn

			1^2+1=2 x=1, x<2*1 and 2 and 3 twin primes so a(1)=2
2^2+1=5 x=1, x<2*2 and 5 and 7 twin primes so a(2)=5
3^2+2=11 x=2, x<2*3 and 11 and 13 twin primes so a(3)=11

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A091592.

A192870 The maximum integer M such that there are no prime n-tuplets of any possible pattern between M^2 and (M+1)^2, or -1 if no such maximum M exists.

Original entry on oeis.org

0, 122, 3113, 719377, 15467683

Offset: 1

Alexei Kourbatov, Jul 11 2011

All terms are conjectural. A prime n-tuplet is defined as the densest permissible prime constellation containing n primes. The term a(2) corresponds to twin primes, a(3) to prime triplets, a(4) to prime quadruplets, etc. Extensive computational evidence suggests that these terms are valid. However, there is no proof that the greatest integer M exists - not even for a subset of values of n. If one could find a constructive existence proof, then Twin Prime Conjecture as well as Legendre's Conjecture would require just a trivial additional step. - Edited by Hugo Pfoertner, Sep 15 2021
Note that, for some n, a prime (n+1)-tuple must include a prime n-tuple; e.g., prime quadruplets include prime triples. Thus, if any term is -1,  subsequent terms may be -1, too. - Franklin T. Adams-Watters and Alexei Kourbatov, Jul 14 2011
However, for other n, a prime (n+1)-tuple does NOT include a prime n-tuple; e.g. 7-tuples {p, p + 2, p + 6, p + 8, p + 12, p + 18, p + 20} do not contain 6-tuples {p-4, p, p + 2, p + 6, p + 8, p + 12}; see List of all possible patterns of prime k-tuplets by Tony Forbes.
Assuming the Hardy-Littlewood k-tuple conjecture, the average distance between k-tuples grows slower than the distance between consecutive squares. This is an indication (but not a proof) that the maximum integer M in this sequence does exist for all n.
a(6) > 3005845357, because there is a gap of size 7191214380 between consecutive sextuplets, enclosing in its interior the interval between the two squares 3005845357^2 and 3005845358^2, found with a fast sieving program provided by Martin Raab, i.e., 9035106309825245467 < 3005845357^2 = 9035106310198457449 < 3005845358^2 = 9035106316210148164 < 9035106317016459847. Statistical considerations based on the observed scatter of maximum gap sizes (see A200503 and data provided at N. Luhn link) suggest a range of 3*10^9 < a(6) < 6*10^9. In a private communication Martin Raab provided the following estimates for the order of magnitude of the next terms: a(7) ~= 1.1*10^11, a(8) ~= 1.6*10^12, a(9) ~= 6*10^13, a(10) ~= 3*10^16. - Hugo Pfoertner, Jul 31 2023, corrected Oct 23 2023

			The term a(4)=719377 means that there are no prime quadruplets between 719377^2 and 719378^2, but there are prime quadruplets between m^2 and (m+1)^2 for m > 719377.

Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242, 2013 and J. Int. Seq. 16 (2013) #13.5.2
Norman Luhn, Record Gaps Between Prime Sextuplets, up to 10^17
Eric W. Weisstein, k-Tuple Conjecture

Cf. A091592: Numbers n such that there are no twin primes between n^2 and (n+1)^2; A008407: Minimal width of prime n-tuplet.

Cf. A020497, A200503.

First term, 0, added and offset changed by Zak Seidov, Jul 11 2011

Clarification regarding patterns in the title added by Hugo Pfoertner, Sep 15 2021

A099154 Largest number k such that the interval [k^2,(k+1)^2] contains not more than n pairs of twin primes.

Original entry on oeis.org

122, 213, 502, 545, 922, 950, 749, 1098, 1330, 1450, 1634, 1623, 2135, 2110, 2177, 2244, 2760, 2413, 2556, 3280, 3454, 3211, 3740, 3540, 4104, 4096, 4391, 4457, 4592, 5309, 4758, 5720, 5747, 5295, 5902, 5456, 5920, 6395, 5810, 7007, 7109, 7450, 7540

Offset: 0

Hugo Pfoertner, Sep 30 2004

nonn

The terms of this sequence are conjectural, even under the twin prime conjecture.

			a(1)=213 because the interval [213^2,214^2]=[45369,45796] contains one pair of twin primes (45587,45589) whereas all higher intervals are conjectured to contain at least two pairs of twin primes.
The interval [122^2,123^2]=[A091592(11)^2,(A091592(11)+1)^2] is conjectured to be the last interval between two consecutive squares containing no twin primes.

Eric Weisstein's World of Mathematics, Twin Primes.
Eric Weisstein's World of Mathematics, Twin Prime Conjecture.

Cf. A091591 number of pairs of twin primes between n^2 and (n+1)^2, A091592 numbers n such that there are no twin primes between n^2 and (n+1)^2, A014574.

A246826 Numbers n such that there is no prime of a prime twin pair between n^2 + n and n^2 + 3*n + 2.

Original entry on oeis.org

0, 10, 26, 30, 36, 136, 156, 433

Offset: 1

Pierre CAMI, Sep 04 2014

nonn

No more values for n = 434 to 45140.
Conjecture: the sequence is finite and given in full.
a(9), if it exists, is greater than 10^5. - Derek Orr, Sep 19 2014

			n = 0 only 1 between 0 and 2 so a(1) = 0.
n = 1 between 2 and 6, 3 is the first of twin pair 3, 5.
For n = 2 to 9 always at least one prime of a twin pair between n^2 + n and n^2 + 3*n + 2.
n = 10 no prime of a twin pair between 110 and 132 so a(2) = 10.

Cf. A091592, A108309.

a(n)=forprime(p=n^2+n,n^2+3*n+2,if(precprime(p-1)==p-2||nextprime(p+1)==p+2,return(0)));return(1)
n=0;while(n<10^5,if(a(n),print1(n,", "));n++) \\ Derek Orr, Sep 19 2014

A276192 Numbers n such that there is no twin prime pair between A000217(n) and A000217(n+1) (n > 0).

Original entry on oeis.org

1, 3, 6, 9, 12, 15, 17, 26, 27, 30, 32, 36, 37, 38, 42, 43, 48, 51, 55, 65, 69, 75, 77, 108, 123, 131, 134, 149, 161, 172, 175, 221, 229, 345, 353, 613

Offset: 1

Altug Alkan, Aug 24 2016

Numbers n such that there is no pair of twin primes p, p+2 with n*(n+1)/2 <= p < p+2 < (n+1)*(n+2)/2.
Number of twin prime pairs between A000217(n) and A000217(n+1) are 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 2, 1, 0, 1, 0, 1, 2, 2, 0, 0, 0, 1, 3, 1, 0, 0, 2, 2, 1, 1, 0, 1, 4, 0, 1, 2, 1, 0, 2, 2, 1, 1, 2, 2, 1, 1, 5, 0, 2, 2, 1, 0, 1, 1, 2, 2, 2, 0, 2, 0, 1, 1, 3, 4, 2, 3, ...
Probably the sequence is finite, and a(36)=613 is the last term. If a(37) exists, then a(37)>10000. - Andrey Zabolotskiy, Aug 24 2016
a(37) > 10^8. - Dana Jacobsen, Aug 29 2016

			3 is a term because there is no twin prime pair between A000217(3) = 6 and A000217(4) = 10, even though 7 is one of a prime pair and between 6 and 10, 5 isn't so the pair doesn't exclude 3.

Cf. A000217, A001097, A091592.

t[n_] := n(n+1)/2;
is[n_] := !Or@@Table[PrimeQ[k] && PrimeQ[k+2], {k, t[n], t[n+1]-3}];
Select[Range[700], is] (* Andrey Zabolotskiy, Aug 24 2016 *)

use ntheory ":all"; sub is_a276192 { my $n=shift; my $t=($n*$n+$n)>>1; twin_prime_count($t,$t+$n+1-2) == 0; } # Dana Jacobsen, Aug 29 2016

use ntheory ":all"; sub is_a276192 { my($n,$t,$e,$p,$prev)=(shift); $t = ($n*$n+$n)>>1; $e=$t+$n+1-2; $p = next_prime($t-1); $prev = next_prime($p); ($prev, $p) = ($p, next_prime($p)) while ($p-$prev) != 2;  $prev > $e; }    my $n=1; for (1..36) { $n++ until is_a276192($n); say "$ ",$n++; } # _Dana Jacobsen, Aug 29 2016

cp's OEIS Frontend

A091591 Number of pairs of twin primes between n^2 and (n+1)^2.

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A138205 Least number k such that n^2+k and n^2+k+2 are the smallest twin primes between squares n^2 and (n+1)^2, or 0 if there is no such k.

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A143509 First of a pair of twin primes of the form n^2+x with x<2*n, or 0 if no such prime exist.

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A192870 The maximum integer M such that there are no prime n-tuplets of any possible pattern between M^2 and (M+1)^2, or -1 if no such maximum M exists.

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A099154 Largest number k such that the interval [k^2,(k+1)^2] contains not more than n pairs of twin primes.

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A246826 Numbers n such that there is no prime of a prime twin pair between n^2 + n and n^2 + 3*n + 2.

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A276192 Numbers n such that there is no twin prime pair between A000217(n) and A000217(n+1) (n > 0).

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