A076077 Group the natural numbers so that the n-th group contains n numbers, the k-th number in a group is a multiple of k and the sum of the group is prime: (2), (1, 4), (3, 8, 6), (5, 10, 12, 16), (7, 14, 9, 24, 25), ...

2, 1, 4, 3, 8, 6, 5, 10, 12, 16, 7, 14, 9, 24, 25, 11, 18, 15, 20, 45, 30, 13, 22, 21, 28, 35, 36, 42, 17, 26, 27, 32, 40, 48, 49, 72, 19, 34, 33, 44, 50, 54, 56, 80, 63, 23, 38, 39, 52, 55, 60, 70, 64, 90, 110, 29, 46, 51, 68, 65, 66, 77, 88, 81, 120, 132, 31, 58, 57, 76, 75

Offset: 1

Author

Amarnath Murthy, Oct 05 2002

Keywords

• nonn
• tabl

Comments

For 1 <= k <= n-2, T(n, k) is the smallest multiple of k not already used. T(n, n-1) must be chosen so that T(n, 1)+...+T(n, n-1) is relatively prime to n. - David Wasserman, Mar 24 2005

Examples

			  2,
  1,  4,
  3,  8,  6,
  5, 10, 12, 16,
  7, 14,  9, 24, 25,
 11, 18, 15, 20, 45, 30,
 13, 22, 21, 28, 35, 36, 42,
 17, 26, 27, 32, 40, 48, 49, 72,
 19, 34, 33, 44, 50, 54, 56, 80, 63,
 23, 38, 39, 52, 55, 60, 70, 64, 90,110,
 29, 46, 51, 68, 65, 66, 77, 88, 81,120,132,
		

Crossrefs

Cf. A076074 (first column), A076075 (diagonal), A076076 (row sums).

Programs

Maple

A076077 := proc(n,k)
    option remember ;
    local a,prev,nprev,kprev,psum;
    for a from k by k do
        prev := false ;
        for nprev from 1 to n-1 do
            for kprev from 1 to nprev do
                if procname(nprev,kprev) = a then
                    prev := true ;
                end if;
            end do:
        end do:
        psum :=0 ;
        for kprev from 1 to k-1 do
            psum := psum+procname(n,kprev) ;
            if procname(n,kprev) = a then
                prev := true ;
            end if;
        end do:
        if not prev then
            if k = n-1 then
                if igcd(a + psum,n) = 1 then
                    return a;
                end if;
            elif k = n then
                if isprime(a + psum) then
                    return a;
                end if;
            else
                return a;
            end if;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 08 2025

Extensions

More terms from David Wasserman, Mar 24 2005