A084161 -id:A084161 - cp's OEIS frontend

A084162 a(n) is the length of the gap in sequence A084161.

3, 8, 12, 16, 24, 32, 48, 56, 60, 68, 72, 88, 108, 128, 148, 152, 200, 224, 240, 248, 252, 260, 272, 280, 324, 360, 420, 444, 460, 516, 520, 540, 628, 684, 696, 716, 720, 744, 800, 884, 960, 1044, 1084

Offset: 0

Sven Simon, May 17 2003

nonn

First occurrence maximum gaps in sequence A002313 (real primes with corresponding complex primes).
From Alexei Kourbatov, Feb 16 2016: (Start)
Dirichlet's theorem on arithmetic progressions and GRH suggest that average gaps between primes of the form 4k + 1 below x are about phi(4)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(4)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(4)=2.
Conjecture: a(n) < phi(4)*log^2(A268963(n)); A268963 are the end-of-gap primes.
(End)
Conjecture: a(n) < phi(4)*n^2 for all n > 2. (Note the starting offset 0.) - Alexei Kourbatov, Aug 12 2017

			a(3) = 16: There are no primes p = 1 mod 4 between 73 and 89, this gap is the largest up to 89, the gap size is 16.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov, Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.

Cf. A002313, A084160, A084161 (start of gap), A268963 (end of gap); A268799, A268925, A268928.

Reap[Print[3]; Sow[3]; r = 0; p = 5; For[q = 7, q < 10^7, q = NextPrime[q], If[Mod[q, 4] == 3, Continue[]]; g = q - p; If[g > r, r = g; Print[g] Sow[g]]; p = q]][[2, 1]] (* Jean-François Alcover, Feb 20 2019, from PARI *)

print1(3); r=0; p=5; forprime(q=7, 1e7, if(q%4==3, next); g=q-p; if(g>r, r=g; print1(", "g)); p=q)

A084160 First occurrence prime gaps of the primes in sequence A002313 (Real primes with corresponding complex primes). a(0) = 2 with length of gap 3. For n>0 the size of the gap in the sequence is 4n, a(n) is the starting prime of the gap.

Original entry on oeis.org

2, 13, 5, 17, 73, 293, 113, 1153, 197, 2557, 1321, 1553, 461, 2161, 1493, 1801, 10993, 9533, 15661, 27817, 76001, 24593, 16741, 40709, 53453, 58789, 62297, 33181, 256189, 110321, 112757, 344497, 39581, 138661, 269761, 448421, 78989, 50593

Offset: 0

Sven Simon, May 17 2003

nonn

Real primes 2,5,13,17,29,37,... have a unique representation as sum of two squares. Values larger 2 are the primes p with p = 1 mod 4. This is sequence A002313. If p = x^2 + y^2, the corresponding complex prime is x+y*i

			a(3) = 17 because the next prime in sequence A002313 is 29, the size of the gap is 3*4 = 12.

Handbook of First Complex Prime Numbers, Part1+2 Ervand Kogbetliantz and Alice Krikorian, Gordon and Breach, 1971

Cf. A002313, A084161.

A268799 Record (maximal) gaps between primes of the form 4k + 3.

Original entry on oeis.org

4, 8, 12, 20, 24, 36, 40, 56, 60, 64, 68, 112, 120, 132, 144, 156, 168, 176, 184, 200, 240, 256, 272, 280, 296, 356, 396, 444, 452, 480, 532, 616, 620, 672, 692, 708, 840, 864, 896, 916, 1004

Offset: 1

Alexei Kourbatov, Feb 13 2016

nonn

Dirichlet's theorem on arithmetic progressions and GRH suggest that average gaps between primes of the form 4k + 3 below x are about phi(4)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(4)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(4)=2.
Conjecture: a(n) < phi(4)*log^2(A268801(n)) almost always.
Conjecture: a(n) < phi(4)*n^2 for all n>2. - Alexei Kourbatov, Aug 12 2017

			The first two primes of the form 4k+3 are 3 and 7, so a(1)=7-3=4. The next prime of this form is 11; the gap 11-7 is not a record so no term is added to the sequence. The next prime of this form is 19; the gap 19-11=8 is a new record, so a(2)=8.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A002145, A084161.

Corresponding primes: A268800 (lower ends), A268801 (upper ends).

re = 0; s = 3; Reap[For[p = 7, p < 10^8, p = NextPrime[p], If[Mod[p, 4] != 3, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* Jean-François Alcover, Dec 12 2018, from PARI *)

re=0; s=3; forprime(p=7, 1e8, if(p%4!=3, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

A268800 Primes 4k + 3 preceding the maximal gaps in A268799.

Original entry on oeis.org

3, 11, 31, 83, 283, 383, 1327, 2591, 7351, 7759, 11171, 11587, 31391, 46919, 147919, 288023, 360611, 425603, 507163, 666203, 1414703, 2198887, 3358151, 9287659, 11512547, 11648531, 24315047, 42453823, 145554779, 161720147, 184007671, 766668811

Offset: 1

Alexei Kourbatov, Feb 13 2016

nonn

Subsequence of A002145.

A268799 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 4k+3 are 3 and 7, so a(1)=3. The next prime of this form is 11; the gap 11-7 is not a record so no term is added to the sequence. The next prime of this form is 19; the gap 19-11=8 is a new record so a(2)=11.

Alexei Kourbatov, Table of n, a(n) for n = 1..41
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A002145, A084161, A268799, A268801.

re=0; s=3; forprime(p=7, 1e8, if(p%4!=3, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

A268801 Primes 4k + 3 at the end of the maximal gaps in A268799.

Original entry on oeis.org

7, 19, 43, 103, 307, 419, 1367, 2647, 7411, 7823, 11239, 11699, 31511, 47051, 148063, 288179, 360779, 425779, 507347, 666403, 1414943, 2199143, 3358423, 9287939, 11512843, 11648887, 24315443, 42454267, 145555231, 161720627, 184008203, 766669427

Offset: 1

Alexei Kourbatov, Feb 13 2016

nonn

Subsequence of A002145.

A268799 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 4k+3 are 3 and 7, so a(1)=7. The next prime of this form is 11; the gap 11-7 is not a record so no term is added to the sequence. The next prime of this form is 19; the gap 19-11=8 is a new record so a(2)=19.

Alexei Kourbatov, Table of n, a(n) for n = 1..41
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A002145, A084161, A268799, A268800.

re=0; s=3; forprime(p=7, 1e8, if(p%4!=3, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

A268963 Primes 4k+1 at the end of the maximal gaps in A084162.

Original entry on oeis.org

5, 13, 29, 89, 137, 229, 509, 1549, 1861, 9601, 15733, 16829, 33289, 39709, 50741, 180949, 183289, 1562053, 1638053, 2244157, 4469141, 4874977, 7856713, 10087481, 12021353, 12214273, 18227081, 148364081, 292182557, 320262769, 468214457, 727335397, 869766761

Offset: 0

Alexei Kourbatov, Feb 16 2016

nonn

Subsequence of A002144.

A084161 lists the primes preceding the maximal gaps, and A084162 lists the corresponding gap sizes. See more comments there.

			a(3) = 89: There are no primes p = 1 mod 4 between 73 and 89, this gap is the largest up to 89, the gap size is 16.

Alexei Kourbatov, Table of n, a(n) for n = 0..42

Cf. A002144, A002313, A084161, A084162.

Reap[Print[5]; Sow[5]; r = 0; p = 5; For[q = 7, q < 10^7, q = NextPrime[q], If[Mod[q, 4] == 3, Continue[]]; g = q - p; If[g > r, r = g; Print[q] Sow[q]]; p = q]][[2, 1]] (* Jean-François Alcover, Feb 20 2019, from PARI *)

print1(5); r=0; p=5; forprime(q=7, 1e9, if(q%4==3, next); g=q-p; if(g>r, r=g; print1(", "q)); p=q)

a(n) = A084161(n) + A084162(n)

cp's OEIS Frontend

A084162 a(n) is the length of the gap in sequence A084161.

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A084160 First occurrence prime gaps of the primes in sequence A002313 (Real primes with corresponding complex primes). a(0) = 2 with length of gap 3. For n>0 the size of the gap in the sequence is 4n, a(n) is the starting prime of the gap.

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A268799 Record (maximal) gaps between primes of the form 4k + 3.

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A268800 Primes 4k + 3 preceding the maximal gaps in A268799.

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A268801 Primes 4k + 3 at the end of the maximal gaps in A268799.

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A268963 Primes 4k+1 at the end of the maximal gaps in A084162.

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