A109653 - OEIS

%I #10 Nov 05 2024 10:30:54

%S 2,5,12,23,36,53,72,101,132,169,210,253,300,359,420,487,558,631,710,

%T 793,882,979,1082,1189,1298,1411,1538,1669,1806,1945,2094,2245,2402,

%U 2565,2732,2905,3084,3265,3456,3649,3846,4045,4256,4479,4706,4935,5168,5407

%N Sequence and first differences include all prime numbers exactly once.

%C Sequence and first differences:

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

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

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

%p A109653diff :=proc(n)

%p     option remember ;

%p     if n = 2 then

%p         3;

%p     else

%p         for pidx from 1 do

%p             fnd := false;

%p             p := ithprime(pidx) ;

%p             for i from 2 to n-1 do

%p                 if procname(i) = p then

%p                     fnd := true;

%p                 end if;

%p             end do:

%p             for i from 2 to n do

%p                 if A109653(i) = p then

%p                     fnd := true;

%p                 end if;

%p             end do:

%p             if not fnd then

%p                 return p;

%p             end if;

%p         end do:

%p     end if;

%p end proc:

%p A109653 :=proc(n)

%p     if n = 2 then

%p         2 ;

%p     else

%p         procname(n-1)+A109653diff(n-1) ;

%p     end if;

%p end proc:

%p seq(A109653(n),n=2..80) ; # _R. J. Mathar_, Nov 05 2024

%t 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_ *)

%Y Cf. A247657

%K base,easy,nonn

%O 2,1

%A _Eric Angelini_, Aug 30 2005

%E More terms from _Robert G. Wilson v_, Sep 28 2005