A202282 -id:A202282 - cp's OEIS frontend

A027569 Initial members of prime decaplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26, p+30, p+32).

11, 33081664151, 83122625471, 294920291201, 573459229151, 663903555851, 688697679401, 730121110331, 1044815397161, 1089869189021, 1108671297731, 1235039237891, 1291458592421, 1738278660731

Offset: 1

Warut Roonguthai

nonn

All terms are congruent to 11 (modulo 210). - Matt C. Anderson, May 28 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 A027569, zip compressed (7.93 MB) (2021)

Cf. A027570, A202281, A202282.

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;
# begin initialization section
p := [0, 2, 6, 8, 12, 18, 20, 26, 30, 32]; o := [1271, 1691]; m := 2310;
# end initialization section
with(ArrayTools); os := Size(o, 2); ps := Size(p, 2);
loopstop := 10^11; loopstart := 0;
print(11);
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 30 2015

is(n)=isprime(n) && isprime(n+2) && isprime(n+6) && isprime(n+8) && isprime(n+12) && isprime(n+18) && isprime(n+20) && isprime(n+26) && isprime(n+30) && isprime(n+32)
v=primes(10); t=1; forprime(p=31,1e11, v[t]=p; t=(t%10)+1; if(p-v[t]==32 && is(v[t]), print1(v[t]", "))) \\ Charles R Greathouse IV, May 20 2015

use ntheory ":all"; say for sieve_prime_cluster(1,1e13, 2,6,8,12,18,20,26,30,32); # Dana Jacobsen, Sep 30 2015

A213647 Initial members of prime 11-tuplets: primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36) are all prime.

Original entry on oeis.org

11, 7908189600581, 10527733922591, 12640876669691, 38545620633251, 43564522846961, 60268613366231, 60596839933361, 71431649320301, 79405799458871, 109319665100531, 153467532929981, 171316998238271, 216585060731771, 254583955361621, 259685796605351, 268349524548221

Offset: 1

Matt C. Anderson, Jun 17 2012

nonn

0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 are the first terms of A135311.
All terms are congruent to 11 (modulo 210). - Zak Seidov, Sep 15 2014
Subsequence of A202282. - Zak Seidov, Sep 15 2014
All terms, except the first one, are congruent to 1271 (modulo 2310). - Matt C. Anderson, May 29 2015

Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..6800 (first 100 terms from Matt C. Anderson)
Norman Luhn, Table of n, a(n) for n = 1..100000

Cf. A135311, A202282.

use ntheory ":all"; say for sieve_prime_cluster(1,1e14, 2,6,8,12,18,20,26,30,32,36); # Dana Jacobsen, Oct 01 2015

a(89) corrected by Dana Jacobsen, Oct 01 2015

A202281 Record (maximal) gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32).

Original entry on oeis.org

33081664140, 50040961320, 211797665730, 278538937950, 314694286830, 446820068310, 589320949140, 1135263664920, 1154348695500, 1280949740070, 1340804150070, 1458168320490, 1539906870810, 1858581264540, 2590180927950, 3182865274050, 4949076176310, 5719502339670

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 (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)).
A202282 lists initial primes in decuplets (p+0,2,6,8,12,18,20,26,30,32) preceding the maximal gaps.

			The gap of 33081664140 after the first decuplet starting at p=11 is the term a(1). The next three gaps of 50040961320, 211797665730, 278538937950 form an increasing sequence, each setting a new record; therefore each of these gaps is in the sequence, as a(2), a(3), a(4). The next gap is not a record, so it is not in this sequence.

Norman Luhn, Table of n, a(n) for n = 1..54 (terms 1..32 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.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A027569 (prime decuplets p+0,2,6,8,12,18,20,26,30,32), A202282, A202361, A113274, A113404, A200503, A201596, A201598, A201062, A201073, A201051, A201251

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 $gap" if ++$i > 0; $max=$gap; } $l=$; } # Dana Jacobsen, Oct 08 2015

(1) Upper bound: gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32) 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.

A202362 Initial prime in prime decuplets (p+0,2,6,12,14,20,24,26,30,32) preceding the maximal gaps in A202361.

Original entry on oeis.org

9853497737, 22741837817, 242360943257, 1418575498577, 4396774576277, 8639103445097, 11105292314087, 12728490626207, 119057768524127, 226608256438997, 581653272077387, 896217252921227, 987041423819807, 1408999953009347, 1419018243046487, 2189095026865907

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. Maximal gaps between decuplets of this type are listed in A202361; see more comments there.

			The gap of 12102794130 between the very first decuplets starting at p=9853497737 and p=21956291867 means that the initial term is a(1)=9853497737.
The next gap after the decuplet starting at p=21956291867 is smaller, so it does not contribute to 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)=22741837817.

Norman Luhn, Table of n, a(n) for n = 1..44 (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
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A027570 (prime decuplets p+0,2,6,12,14,20,24,26,30,32), A202281, A202282, A202361.

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 $l" if ++$i > 0; $max=$gap; } $l=$; } # Dana Jacobsen, Oct 09 2015

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A027569 Initial members of prime decaplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26, p+30, p+32).

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A213647 Initial members of prime 11-tuplets: primes p such that p + (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36) are all prime.

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A202281 Record (maximal) gaps between prime decuplets (p+0,2,6,8,12,18,20,26,30,32).

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A202362 Initial prime in prime decuplets (p+0,2,6,12,14,20,24,26,30,32) preceding the maximal gaps in A202361.

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