A027570 -id:A027570 - cp's OEIS frontend

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

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.

A186634 Irregular triangle, read by rows, giving dense patterns of n primes.

Original entry on oeis.org

0, 2, 0, 2, 6, 0, 4, 6, 0, 2, 6, 8, 0, 2, 6, 8, 12, 0, 4, 6, 10, 12, 0, 4, 6, 10, 12, 16, 0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20, 0, 2, 6, 8, 12, 18, 20, 26, 0, 2, 6, 12, 14, 20, 24, 26, 0, 6, 8, 14, 18, 20, 24, 26, 0, 2, 6, 8, 12, 18, 20, 26, 30, 0, 2, 6, 12, 14, 20, 24, 26, 30, 0, 4, 6, 10, 16, 18, 24, 28, 30, 0, 4, 10, 12, 18, 22, 24, 28, 30, 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 0, 2, 6, 12, 14, 20, 24, 26, 30, 32

Offset: 2

T. D. Noe, Feb 24 2011

The first pattern (0,2) is for twin primes (p,p+2). Row n contains A083409(n) patterns, each one consisting of 0 followed by n-1 terms. In each row the patterns are in lexicographic order.

These numbers (in a slightly different order) appear in Table 1 of the paper by Tony Forbes. Sequence A186702 gives the least prime starting a given pattern.

			The irregular triangle begins:
0, 2
0, 2, 6, 0, 4, 6
0, 2, 6, 8
0, 2, 6, 8, 12, 0, 4, 6, 10, 12
0, 4, 6, 10, 12, 16
0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20

T. D. Noe, Rows n = 2..20, flattened (from Forbes)
Thomas J. Engelsma, Permissible Patterns
Tony Forbes, Prime clusters and Cunningham chains, Math. Comp. 68 (1999), 1739-1747.
Tony Forbes, Smallest Prime k-tuples
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A020497, A008407, A186702, and sequences created by these patterns: A001359, A022004, A022005, A007530, A022006, A022007, A022008, A022009, A022010, A022011, A022012, A022013, A022545, A022546, A022547, A022548, A027569, A027570.

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

A281256 Runs of consecutive integers in A270877, which is produced by a decaying trapezoidal modification of the sieve of Eratosthenes.

Original entry on oeis.org

8, 13, 1, 19, 16, 4, 32, 64, 22, 49, 34, 166, 27, 71, 38, 44, 172, 59, 302, 1984, 46771, 56, 178, 94, 346, 4925, 59492, 188357, 68, 205, 352, 617, 7408, 113492, 371918, 881212, 80, 211, 382, 939, 9110, 114602, 964583, 6671161, 24365591, 89, 214, 581, 1011, 11090, 207938, 1008362

Offset: 1

Peter Munn, Jan 18 2017

Square table T, read by ascending antidiagonals, where T(n,m) gives the least integer in the n-th occurrence of a run of exactly m consecutive integers in the ordered sequence A270877.
A270877 is sifted from the positive integers by modifying the sieve of Eratosthenes: instead of eliminating integers that would enumerate a rectangular area dot pattern with one side held at a constant length (equal to each surviving integer in turn), the sieve eliminates those enumerating a trapezoidal area dot pattern with the constant length being the trapezoid's longest side. Given this geometric relationship, it is considered worth looking for qualities that A270877 may have in common with the sequence of primes, potentially influenced by related causes such as the effect of prime factors on A270877.
The columns of this sequence, listing the runs of m consecutive integers within A270877, merit comparative examination with equivalent sequences for prime k-tuples. For m=5, the notably larger ratio between T(1,5) and T(2,5) resembles early large ratio gaps in the occurrence sequences of k-tuples such as A022008 (sextuples), whereas columns m<5 are more comparable with those for shorter k-tuples such as A001359 (twin primes) and A007530 (quadruples), each having a relatively low-valued first term (less than 60) and without such a large ratio gap. In comparison, the columns for runs m>5 appear more like the sequences for some longer k-tuples such as A027570 (a 10-tuple sequence). Row 1 merits comparative examination with A186702 for primes.
The author conjectures that T(n,m) exists for all n>=1, m>=1.

			4, 5 and 6 occur in A270877, but 3 and 7 do not. This is the first run of exactly 3 consecutive integers in A270877, so T(1,3) = 4.
Square table T(n,m) begins:
   8,   1,   4,   49,    38,  46771,  188357,   881212, ...
  13,  16,  22,   71,  1984,  59492,  371918,  6671161, ...
  19,  64,  27,  302,  4925, 113492,  964583,  8799769, ...
  32, 166,  59,  346,  7408, 114602, 1008362, 13579777, ...
  34, 172,  94,  617,  9110, 207938, 1094293, 14874616, ...
  44, 178, 352,  939, 11090, 291712, 1156214, 15974752, ...
  56, 205, 382, 1011, 13007, 323716, 1239046, 20585962, ...
  68, 211, 581, 1080, 13216, 429915, 1433918, 20745838, ...
  80, 214, 599, 1091, 14710, 442807, 1702694, 24321313, ...
  89, 223, 624, 1151, 15052, 457220, 1712927, 25634557, ...

This is an analysis of A270877.

Cf. A001359, A007530, A022008, A027570, A186702.

A350830 Number of prime 10-tuples (or decaplets) with initial member (A257127) between 10^(n-1) and 10^n.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 4, 15, 81, 357, 1685, 8256

Offset: 1

M. F. Hasler, Mar 01 2022

"Between 10^(n-1) and 10^n" is equivalent to saying "with n (decimal) digits".
A prime 10-tuple (or decaplet) is a sequence of 9 consecutive primes (p1, ..., p10) of minimum possible diameter p10 - p1 = 32.
Terms a(1)-a(16) computed from b-files a(1..10^4) for A027569 and A027570. Using Luhn's data (cf. LINKS) one can obtain a(18) and a(19).
So far the last term of all the decaplets has the same number of digits as the initial term.

			a(2) = 1 because 11 is the only two-digit prime to start a prime decaplet, i.e., member of A257127.
a(n) = 0 for all other n < 10 because the next larger prime decaplet is made of 10-digit primes, A257127(2) = 9853497737 and successors.
a(10) = 1 because there is only one prime decaplet made of 10-digit primes.
a(11) = 4 because there are only four terms in A257127 (for indices n = 3..6) which have 11 digits.

Norman Luhn, The big database of "The smallest prime k-tuplets", section "10-uplets": up to 10^20 as of March 2022.

Cf. A257127 (initial members p of prime 9-tuples (p, ..., p+32)), A027569, A027570 (idem, specifically for each of the two possible patterns).

Cf. A350825 - A350829: similar for quintuples, sextuples, septuples, octuples and 9-tuples.

(D(v)=v[^1]-v[^-1])( [setsearch(A257127,10^n,1) | n<-[0..16]] ) \\ where A257127 is a vector of at least 10400 terms of that sequence.

cp's OEIS Frontend

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

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A186634 Irregular triangle, read by rows, giving dense patterns of n primes.

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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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A281256 Runs of consecutive integers in A270877, which is produced by a decaying trapezoidal modification of the sieve of Eratosthenes.

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A350830 Number of prime 10-tuples (or decaplets) with initial member (A257127) between 10^(n-1) and 10^n.

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