A109653 Sequence and first differences include all prime numbers exactly once.

2, 5, 12, 23, 36, 53, 72, 101, 132, 169, 210, 253, 300, 359, 420, 487, 558, 631, 710, 793, 882, 979, 1082, 1189, 1298, 1411, 1538, 1669, 1806, 1945, 2094, 2245, 2402, 2565, 2732, 2905, 3084, 3265, 3456, 3649, 3846, 4045, 4256, 4479, 4706, 4935, 5168, 5407

Offset: 2

Author

Eric Angelini, Aug 30 2005

Keywords

• base
• easy
• nonn

Comments

Sequence and first differences:

2 5 12 23 36 53 72 101 132 169 210 253 300 359 420...

.3.7.11.13.17.19.29...31..37..41..43..47..59..61...

Examples

			All prime numbers appear once and only once, either in the sequence itself or in the first differences.
		

Crossrefs

Cf. A247657

Programs

Maple

A109653diff :=proc(n)
    option remember ;
    if n = 2 then
        3;
    else
        for pidx from 1 do
            fnd := false;
            p := ithprime(pidx) ;
            for i from 2 to n-1 do
                if procname(i) = p then
                    fnd := true;
                end if;
            end do:
            for i from 2 to n do
                if A109653(i) = p then
                    fnd := true;
                end if;
            end do:
            if not fnd then
                return p;
            end if;
        end do:
    end if;
end proc:
A109653 :=proc(n)
    if n = 2 then
        2 ;
    else
        procname(n-1)+A109653diff(n-1) ;
    end if;
end proc:
seq(A109653(n),n=2..80) ; # R. J. Mathar, Nov 05 2024

Mathematica

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = {1}; d = 3; k = 2; Do[ While[ Position[a, d] != {}, d += 2 ]; k = k + d; d = NextPrim[d]; a = Append[a, k], {n, 47} ]; a (* Robert G. Wilson v *)

Extensions

More terms from Robert G. Wilson v, Sep 28 2005