A088643 -id:A088643 - cp's OEIS frontend

A049478 Record values in A346778.

0, 1, 4, 6, 12, 22, 27, 62, 64, 84, 104, 106, 108, 114, 132, 142, 178, 179, 186, 213, 216, 247, 273, 322, 341, 357, 358, 449, 458, 462, 470, 478, 482, 495, 499, 502, 581, 585, 597, 598, 623, 662, 672, 681, 735, 772, 778, 794, 818, 827, 851, 864

Offset: 1

N. J. A. Sloane

Record values of the least g >= 1 such that the first g terms of row n of A088643 [start with n and always append the largest k < n not yet used such that n+k is a prime] constitute the set {n, n-1, ..., n-g+1}, and the next (i.e., (g+1)-th) term of that row equals n-g. - M. F. Hasler, Aug 04 2021

Cf. A049476, A049477, A088643, A346778.

{print1(m=0); for( n=1, oo, my( r=A088643_row(n)); for( g=1, #r-1, if( Set(r[1..g]) == [n-g+1..n] && r[g+1]==n-g, g > m && print1(", "m=g); break)))} \\ M. F. Hasler, Aug 04 2021

Revised by Sean A. Irvine, Aug 03 2021

A132075 A conjectured permutation of the positive integers such that for every n, a(n) is the largest number among a(1), a(2), ..., a(n) that when added to a(n+1) gives a prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 14, 9, 10, 13, 16, 15, 8, 11, 20, 17, 12, 19, 24, 23, 18, 25, 22, 21, 26, 27, 34, 33, 28, 31, 30, 29, 32, 35, 36, 37, 46, 43, 40, 39, 44, 45, 38, 41, 42, 47, 50, 59, 54, 55, 58, 51, 62, 65, 48, 61, 52, 57, 56, 53, 60, 49, 64, 63, 68, 69, 70, 67, 72, 77, 80, 71, 66, 73, 78, 79, 84, 83, 90, 89, 74, 75, 76, 81, 82, 85, 88, 93, 86, 95, 104, 107, 92

Offset: 1

Paul Boddington, Oct 30 2007, Mar 06 2010

The terms are defined as follows. Start by choosing the initial terms: 1, 2, 3. Then write the rows of table A088643 backwards but always leave off the last three quarters of the terms. This gives: [], [], [], [1], [1], [1], [1], [1, 2], [1, 2], [1, 2], [1, 4], [1, 4, 3,], [1, 4, 3] etc. Then build the sequence up by repeatedly choosing the first such truncated row that extends the terms already chosen. [Edited by Peter Munn, Aug 19 2021]
It is not until the 26th truncated row - [1, 2, 3, 4, 7, 6] - that the initial list is extended at all. It is unclear whether this process can be continued indefinitely, although I have verified by computer that it generates a sequence of at least 2000 terms. Conjecturally: (1) the sequence is infinite, (2) it is the unique sequence containing infinitely many complete rows of table A088643, and (3) for every n > 0 there exists N > 0 such that the first n terms of this sequence are contained in every row of table A088643 from the N-th onwards.
Maybe the idea could be expressed more concisely by defining this sequence as the limit of the reversed rows of A088643? - M. F. Hasler, Aug 04 2021
It seems we do not know of an existence proof for the limit of the reversed rows of A088643. - Peter Munn, Aug 19 2021

Peter Munn, Illustration of the relationship between A088643, this sequence and A255312.

Cf. A088643.

Name edited by Peter Munn, Aug 19 2021

A132163 Triangle read by rows. For row n, start with 1 but from the second term onwards always choose the largest positive integer between 1 and n inclusive that i) has not already appeared in the row ii) gives a prime when added to the previous term. Stop if no such integer can be found.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 1, 4, 3, 2, 5, 1, 6, 5, 2, 3, 4, 1, 6, 7, 4, 3, 2, 5, 1, 6, 7, 4, 3, 8, 5, 2, 1, 6, 7, 4, 9, 8, 5, 2, 3, 1, 10, 9, 8, 5, 6, 7, 4, 3, 2, 1, 10, 9, 8, 11, 6, 7, 4, 3, 2, 5, 1, 12, 11, 8, 9, 10, 7, 6, 5, 2, 3, 4

Offset: 1

Paul Boddington, Nov 04 2007

The following statements are conjectural: 1) The n-th row is always a permutation of 1,...,n. 2) For the even rows, the last term is one less than a prime (so the row gives a solution to the prime circle problem - see A051252). 3) There exists a (unique) sequence b(2), b(3),... with the property that for every n > 1 there is a positive integer N such that every even row of the triangle from the 2N-th onwards ends b(n), ..., b(3), b(2) and every odd row from the (2N - 1)-th onwards ends b(n)+(-1)^n, ..., b(3)-1, b(2)+1. (If the sequence b(n) exists it is probably A132075 without the initial term 1.)

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

This sequence is a variation on A088643.

import Data.List (delete)
a132163_tabl = map a132163_row [1..]
a132163 n k = a132163_row n !! (k-1)
a132163_row n = 1 : f 1 [n, n-1 .. 2] where
   f u vs = g vs where
     g []                            = []
     g (x:xs) | a010051 (x + u) == 1 = x : f x (delete x vs)
              | otherwise            = g xs
-- Reinhard Zumkeller, Jan 05 2013

t[, 1] = 1; t[n, k_] := t[n, k] = For[ j = n, j > 1, j--, If[ PrimeQ[ t[n, k-1] + j] && FreeQ[ Table[ t[n, m], {m, 1, k-1}], j], Return[j] ] ]; Table[ t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 02 2013 *)

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A049478 Record values in A346778.

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A132075 A conjectured permutation of the positive integers such that for every n, a(n) is the largest number among a(1), a(2), ..., a(n) that when added to a(n+1) gives a prime.

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Author

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Comments

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Crossrefs

Extensions

A132163 Triangle read by rows. For row n, start with 1 but from the second term onwards always choose the largest positive integer between 1 and n inclusive that i) has not already appeared in the row ii) gives a prime when added to the previous term. Stop if no such integer can be found.

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Haskell

Mathematica