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A051237
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Lexicographically earliest prime pyramid, read by rows.
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11
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1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 4, 3, 2, 5, 1, 4, 3, 2, 5, 6, 1, 4, 3, 2, 5, 6, 7, 1, 2, 3, 4, 7, 6, 5, 8, 1, 2, 3, 4, 7, 6, 5, 8, 9, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 1, 2, 3, 4, 7, 10, 9, 8, 5, 6, 11, 1, 2, 3, 4, 7, 10, 9, 8, 5, 6, 11, 12, 1, 2, 3, 4, 7, 6, 5, 12, 11, 8, 9, 10, 13, 1, 2, 3, 4, 7, 6, 13, 10
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row n begins with 1, ends with n and sum of any two adjacent entries is prime.
From Daniel Forgues, May 17 and May 18 2011: (Start)
Since the sum of any two adjacent entries is at least 3, the sum is thus an odd prime, which implies that any two consecutive entries have opposite parity.
Since the first and last entries of row n are fixed to 1 and n, we have to find n-2 entries, where ceiling((n-2)/2) are even and floor((n-2)/2) are odd, so for row n the number of arrangements to investigate is
(ceiling((n-2)/2))! * (floor((n-2)/2))! (Cf. A010551(n-2), n >= 2)
The number of ways of arranging row n to get a prime pyramid is given by A036440. List them in lexicographic order and pick the first (earliest) to get row n of lexicographically earliest prime pyramid.
Prime pyramids are also (more fittingly?) called prime triangles. (End)
It appears that the limit of the rows of the lexicographically earliest prime pyramid is A055265 (see comment on that sequence).
Assuming Dickson's conjecture (or the later Hardy-Littlewood Conjecture B), no backtracking is needed: if the first n-2 elements in each row are chosen greedily, a penultimate member can be chosen such that its sums are prime. [Charles R Greathouse IV, May 18, 2011]
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REFERENCES
| R. K. Guy, Unsolved Problems Number Theory, Section C1.
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LINKS
| T. D. Noe, Rows n=1..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Prime Triangle
OEIS Wiki, Prime triangles
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EXAMPLE
| Triangle begins:
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 4, 3, 2, 5;
1, 4, 3, 2, 5, 6;
1, 4, 3, 2, 5, 6, 7;
1, 2, 3, 4, 7, 6, 5, 8;
1, 2, 3, 4, 7, 6, 5, 8, 9;
1, 2, 3, 4, 7, 6, 5, 8, 9, 10;
1, 2, 3, 4, 7, 10, 9, 8, 5, 6, 11;
1, 2, 3, 4, 7, 10, 9, 8, 5, 6, 11, 12;
1, 2, 3, 4, 7, 6, 5, 12, 11, 8, 9, 10, 13;
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MATHEMATICA
| (* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Block[{r = Range@ n}, While[ Union[ PrimeQ[ Plus @@@ Partition[r, 2, 1]]][[1]] == False, r = NextPermutation@ r]; r]; f[1] = 1; Array[f, 13] // Flatten (* RGWv *)
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CROSSREFS
| See A187869 for the concatenation of the numbers for each row.
Cf. A036440, A051239, A055265.
Sequence in context: A183534 A066040 A066019 * A064379 A194973 A195113
Adjacent sequences: A051234 A051235 A051236 * A051238 A051239 A051240
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KEYWORD
| tabf,nice,nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from JudMcCranie(AT)ugaalum.uga.edu.
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