A124939 Prime tetrahedron, read by rows.

1, 1, 1, 2, 1, 1, 4, 1, 6, 5, 1, 1, 10, 1, 12, 7, 1, 16, 3, 8, 1, 1, 18, 1, 22, 9, 1, 28, 13, 24, 1, 30, 11, 20, 17, 1, 1, 36, 1, 40, 19, 1, 42, 25, 34, 1, 46, 15, 14, 23, 1, 52, 21, 26, 27, 32, 1, 1, 58, 1, 60, 29, 1, 66, 31, 48, 1, 70, 33, 38, 35, 1, 72, 37, 64, 39, 44, 1, 78, 49, 54, 43

Offset

1,4

Comments

Each triangular layer of the unique tetrahedron begins with 1, never uses any value other than 1 which has occurred already on this or earlier levels, always uses the least available integer such that the sum of each two consecutive entries is a prime. The number of values of the n-th level is the n-th triangular number A000217(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. The number of values through the n-th level is the n-th tetrahedral number A000292(n) = C(n+2,3) = n(n+1)(n+2)/6.

References

R. K. Guy, Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994.

Kenney, M. J. "Student Math Notes." NCTM News Bulletin. Nov. 1986.

Links

Table of n, a(n) for n=1..82.

Eric Weisstein's World of Mathematics, Prime Triangle.

Formula

a(n) flattens the 3-D table so that level 1 (the apex, with only the value 1) occurs first, then level 2 (with values 1, 1, 2), then level 3 ... and for each level, reads that triangle by rows.

Example

Tetrahedron begins
=================
1
=================
1
1..2
=================
1
1..4
1..6..5
=================
1
1.10
1.12..7
1.16..3..8
=================
1
1.18
1.22..9
1.28.13.24
1.30.11.20.17
=================

Maple

srch := proc(a) local res ; res := 2 ; while true do if isprime(res+op(-1, a)) and not ( res in a ) then RETURN(res) ; fi ; res := res+1 ; od ; end: a := [] ; for lvl from 1 to 10 do for row from 1 to lvl do for col from 1 to row do if col = 1 then anxt := 1 ; else anxt := srch(a) ; fi ; printf("%d, ", anxt) ; a := [op(a), anxt] ; od ; od ; od ; # R. J. Mathar, Jan 13 2007

Crossrefs

Cf. A000040, A000217, A000292, A036440 Number of ways of arranging row n of the Prime Pyramid, A051239, A051237 Lexicographically earliest Prime Pyramid, read by rows.

Sequence in context: A372647 A375023 A021477 * A187800 A340189 A323873

Adjacent sequences: A124936 A124937 A124938 * A124940 A124941 A124942

Keyword

easy,nonn,tabf

Author

Jonathan Vos Post, Nov 13 2006

Extensions

Corrected and extended by R. J. Mathar, Jan 13 2007

Status

approved