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

2,1

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...

Links

Table of n, a(n) for n=2..49.

Example

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

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 *)

Crossrefs

Cf. A247657

Sequence in context: A326762 A116711 A247658 * A115520 A116735 A096376

Adjacent sequences: A109650 A109651 A109652 * A109654 A109655 A109656

Keyword

base,easy,nonn

Author

Eric Angelini, Aug 30 2005

Extensions

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

Status

approved