A186702 -id:A186702 - cp's OEIS frontend

A374000 a(n) = Product_{i=1..m} prime(k + T(n,i)) where k = pi(A186702(n)), T(n,i) is the i-th term in row n of A186634, and m = length of row n of A186634.

15, 385, 1001, 5005, 85085, 323323, 7436429, 955049953, 183698727318433150098859517, 35336848261, 435656388001, 3868985835982814590518552822749329543261, 1448810778701, 20475850236047, 5663533044013, 343523383391078124677551786579090220816600929, 62298863484143

Offset: 1

Michael De Vlieger, Jul 04 2024

nonn

			Let p = A186702 and let T(n,i) be the i-th term in row n of A186634.
a(1) = 15 since p(1) = 3 and row 1 of T is {0, 2}, hence 3 * (3+2) = 3 * 5 = 15.
a(2) = 385 since p(2) = 5 and row 2 of T is {0, 2, 4}, hence 5 * (5+2) * (5+2+4) = 5*7*11 = 385.
Prime decomposition of the first 8 terms.
        a(n)    k  k+m-1  prime decomposition.
----------------------------------------------
         15     2     3    3 *  5
        385     3     5    5 *  7 * 11
       1001     4     6    7 * 11 * 13
       5005     3     6    5 *  7 * 11 * 13
      85085     3     7    5 *  7 * 11 * 13 * 17
     323323     4     8    7 * 11 * 13 * 17 * 19
    7436429     4     9    7 * 11 * 13 * 17 * 19 * 23
  955049953     5    11   11 * 13 * 17 * 19 * 23 * 29 * 31

Cf. A005117, A120944, A186634, A186702.

A214947 Primes p such that p + (0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48) are all prime.

Original entry on oeis.org

186460616596321, 7582919852522851, 31979851757518501, 49357906247864281, 79287805466244211, 85276506263432551, 89309633704415191, 89374633724310001, 98147762882334001, 136667406812471371, 137803293675931951, 152004604862224951, 157168285586497021, 159054409963103491

Offset: 1

Matt C. Anderson, Jul 30 2012

nonn

These are prime 13-tuplets.
All terms congruent to 991 (modulo 2310). - Matt C. Anderson, May 29 2015
All terms congruent to 14851 or 24091 (modulo 30030). - Matt C. Anderson, May 31 2015

Vladimir Vlesycit and Matt C. Anderson and Dana Jacobsen, Table of n, a(n) for n = 1..854 [first 20 terms from Vladimir Vlesycit, first 82 terms from Matt C. Anderson]
Tony Forbes and Norman Luhn, Smallest Prime k-tuplets
Norman Luhn, Table of n, a(n) for n = 1..5579

Cf. A186702.

use ntheory ":all"; say for sieve_prime_cluster(1,10**15, 6,12,16,18,22,28,30,36,40,42,46,48); # Dana Jacobsen, Oct 07 2015

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.

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.

cp's OEIS Frontend

A374000 a(n) = Product_{i=1..m} prime(k + T(n,i)) where k = pi(A186702(n)), T(n,i) is the i-th term in row n of A186634, and m = length of row n of A186634.

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A214947 Primes p such that p + (0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48) are all prime.

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

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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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Examples

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