A082737 Triangle read by rows in which the first row contains a 1 and the n-th row contains n primes not included in the previous rows such that the sum of a row is a perfect square.

1, 2, 7, 3, 5, 17, 11, 13, 19, 101, 23, 29, 31, 37, 241, 41, 43, 47, 53, 59, 157, 61, 67, 71, 73, 79, 83, 191, 89, 97, 103, 107, 109, 113, 127, 1019, 131, 137, 139, 149, 151, 163, 167, 173, 311, 179, 181, 193, 197, 199, 211, 223, 227, 229, 277, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 2689

Offset

1,2

Comments

Another rearrangement of primes and 1.

Links

Table of n, a(n) for n=1..66.

Example

Triangle begins:
   1
   2  7
   3  5 17
  11 13 19 101
  23 29 31  37 241
  ...

Maple

A082737 := proc(nmax) local a, n, r, i, rsum, c, j ; a := [1, 2, 7] ; n := 3 ; i := 1 ; while nops(a)< nmax do r := [] ; for c from 1 to n-1 do while ithprime(i) in a or ithprime(i) in r do i:= i+1 ; od ; r := [op(r), ithprime(i)] ; i:= i+1 ; od ; j := i+1 ; rsum := sum(op(k, r), k=1..nops(r)) ; while not issqr( rsum+ithprime(j)) do j := j+1 ; od ; r := [op(r), ithprime(j)] ; a := [op(a), op(r)] ; n := n+1 ; od ; RETURN(a) ; end: a := A082737(80) : for n from 1 to nops(a) do printf("%d, ", op(n, a)) ; od ; # R. J. Mathar, Nov 12 2006

Mathematica

A082737[nmax_] := Module[{a, n, r, i, rsum, c, j}, a = {1, 2, 7}; n = 3; i = 1; While[Length[a] <= nmax, r = {}; For[c = 1, c <= n-1, c++, While[MemberQ[a, Prime[i]] || MemberQ[r, Prime[i]], i++]; r = Append[r, Prime[i]]; i++]; j = i+1; rsum = Total[r]; While[!IntegerQ@Sqrt[rsum + Prime[j]], j++]; r = Append[r, Prime[j]]; a = Join[a, r]; n++]; Return[a]];
rows = 11;
nmax = rows(rows+1)/2;
tri = A082737[nmax];
T = Table[tri[[(n^2-n+2)/2 ;; n(n+1)/2]], {n, 1, rows}];
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2023, after R. J. Mathar *)

Crossrefs

Main diagonal gives A082738.

Cf. A008578, A082739, A082740.

Sequence in context: A365768 A330883 A075639 * A334971 A104957 A329333

Adjacent sequences: A082734 A082735 A082736 * A082738 A082739 A082740

Keyword

nonn,tabl

Author

Amarnath Murthy, Apr 14 2003

Extensions

More terms from R. J. Mathar, Nov 12 2006

Last row completed by Jean-François Alcover, Jun 22 2023

Status

approved