A202362 -id:A202362 - cp's OEIS frontend

A027570 Initial members of prime decaplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30, p+32).

9853497737, 21956291867, 22741837817, 164444511587, 179590045487, 217999764107, 231255798857, 242360943257, 666413245007, 696391309697, 867132039857, 974275568237, 976136848847, 1002263588297, 1086344116367

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 167 (modulo 210). - Matt C. Anderson, May 29 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 100 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, 1 million terms of A027570, zip compressed (7.94 MB) (2021).

Cf. A027569, A202361, A202362.

a := 1:
for b to 25 do
a := a*ithprime(b):
end do:
a;
# now 'a' is the product of the primes less than 100.
composite_small := proc (n::integer)
description "procedure to determine if n has a prime factor less than 100";
if igcd(2305567963945518424753102147331756070, n) = 1 then return false
else return true;
end if;
end proc:
# so composite_small tests if there are any factors 2 through 97.
#begin initialization section
p := [0, 2, 6, 12, 14, 20, 24, 26, 30, 32];
o := [7517, 10247, 12137, 14447, 14867, 17177, 21377, 24107, 25997, 28727];
m := 30030;
#end initialization section
# implement isprime(m*n+o+p)
with(ArrayTools):
os:=Size(o,2):
ps:=Size(p,2):
#here ps is 10 so a prime constellation of length 10.
loopstop := 10^11:
loopstart := 0:
for n from loopstart to loopstop do
for a to os do
counter := 0; wc := 0; wd := 0;
  while `and`(wd > -10, wd < ps) do
  wd := wd+1;
  if composite_small(m*n+o[a]+p[wd]) = false then wd := wd+1
  else wd := -10 end if;
  end do;
if wd >= 9 then
while `and`(counter >= 0, wc < ps) do
  wc := wc+1;
  if isprime(m*n+o[a]+p[wc]) then counter := counter+1;
  else counter := -1
  end if;
end do;
end if;
if counter = ps then print(m*n+o[a]) end if;
end do:
end do:
# Matt C. Anderson, Apr 15 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e13, 2,6,12,14,20,24,26,30,32); # Dana Jacobsen, Sep 30 2015

A202361 Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32).

Original entry on oeis.org

12102794130, 141702673770, 424052301750, 699699330330, 714303547230, 739544215410, 1623198312120, 2691533434590, 4207848555330, 4936074819480, 5887574660310, 6562654104930, 7205070907650, 8129061524010, 8362548652500, 9741706748970, 9967327212570

Offset: 1

Alexei Kourbatov, Dec 18 2011

nonn

Prime decuplets (p+0,2,6,12,14,20,24,26,30,32) are one of the two types of densest permissible constellations of 10 primes (A027569 and A027570).
Average gaps between prime k-tuples are O(log^k(p)), with k=10 for decuplets, by the Hardy-Littlewood k-tuple conjecture. If a gap is larger than any preceding gap, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps are O(log^11(p)).
A202362 lists initial primes in decuplets (p+0,2,6,12,14,20,24,26,30,32) preceding the maximal gaps.

			The gap of 12102794130 between the very first decuplets starting at p=9853497737 and p=21956291867 is the initial term a(1)=12102794130.
The next gap after the decuplet starting at p=21956291867 is smaller, so it is not in this sequence.
The next gap of 141702673770 between the decuplets at p=22741837817 and p=164444511587 is a new record; therefore the next term is a(2)=141702673770.

Norman Luhn, Table of n, a(n) for n = 1..49 (terms 1..27 from Dana Jacobsen).
Tony Forbes and Norman Luhn, Prime k-tuplets
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 Math. 44, 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A027570 (prime decuplets p+0,2,6,12,14,20,24,26,30,32), A202362, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251, A202281.

use ntheory ":all"; my($i,$l,$max)=(-1,0,0); for (sieve_prime_cluster(1,1e13,2,6,12,14,20,24,26,30,32)) { my $gap=$-$l; if ($gap>$max) { say "$i $gap" if ++$i > 0; $max=$gap; } $l=$; } # Dana Jacobsen, Oct 09 2015

(1) Upper bound: gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32) are smaller than 0.00059*(log p)^11, where p is the prime at the end of the gap.
(2) Estimate for the actual size of maximal gaps near p: max gap = a(log(p/a)-0.2), where a = 0.00059*(log p)^10 is the average gap between 10-tuples near p.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof.

A202282 Initial prime in prime decuplets (p+0,2,6,8,12,18,20,26,30,32) preceding the maximal gaps in A202281.

Original entry on oeis.org

11, 33081664151, 83122625471, 294920291201, 730121110331, 1291458592421, 4700094892301, 6218504101541, 7908189600581, 10527733922591, 21939572224301, 23960929422161, 30491978649941, 46950720918371, 84254447788781, 118565337622001, 124788318636251, 235474768767851

Offset: 1

Alexei Kourbatov, Dec 15 2011

nonn

Prime decuplets (p+0,2,6,8,12,18,20,26,30,32) are one of the two types of densest permissible constellations of 10 primes. Maximal gaps between decuplets of this type are listed in A202281; see more comments there.

			The first four gaps (after the decuplets starting at p=11, 33081664151, 83122625471, 294920291201) form an increasing sequence, with the size of each gap setting a new record. Therefore these values of p are in the sequence, as a(1), a(2), a(3), a(4). The next gap is not a record, so the respective initial prime is not in the sequence.

Norman Luhn, Table of n, a(n) for n = 1..50 (terms 1..33 from Dana Jacobsen).
Tony Forbes and Norman Luhn, Prime k-tuplets
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 Math. 44, 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime k-tuples
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A027569 (prime decuplets p+0,2,6,8,12,18,20,26,30,32), A202281, A202361, A202362.

use ntheory ":all"; my($i,$l,$max)=(-1,0,0); for (sieve_prime_cluster(1,1e13,2,6,8,12,18,20,26,30,32)) { my $gap=$-$l; if ($gap>$max) { say "$i $l" if ++$i > 0; $max=$gap; } $l=$; } # Dana Jacobsen, Oct 09 2015

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A027570 Initial members of prime decaplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30, p+32).

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A202361 Record (maximal) gaps between prime decuplets (p+0,2,6,12,14,20,24,26,30,32).

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A202282 Initial prime in prime decuplets (p+0,2,6,8,12,18,20,26,30,32) preceding the maximal gaps in A202281.

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Perl