A123055 - OEIS

%I #15 Apr 18 2021 02:38:41

%S 1,4,6,25,30,77,84,95,108,125,148,177,208,245,286,329,382,441,502,573,

%T 640,713,792,875,964,1065,1162,1265,1372,1485,1594,1725,1852,1989,

%U 2128,2277,2428,2585,2748,2915,3088,3267,3448,3639,3832,4029,4228,4439,4662

%N Lexicographically earliest sequence of nonprime positive integers whose first differences represent all prime numbers with exactly one appearance of each prime.

%C First differences give the much older sequence A075568. - _N. J. A. Sloane_, Sep 26 2006

%C The n-th prime appears in the sequence of first differences (A075568) not later than at the 2n-th position. To see this it is enough to notice that in the original sequence (excluding the first element) odd and even numbers alternate. Therefore from each odd element m the sequence simply jumps to an even element m+p where p is the smallest previously unused prime. - _Max Alekseyev_, Sep 26 2006

%e The sequence S is constructed like this:

%e Start with "1": S = 1, ...

%e Add the smallest prime not added so far, in order to get a composite:

%e - Can we add 2? No, because 1+2=3 and 3 is prime.

%e - Can we add 3? Yes, because 1+3=4 and 4 is composite.

%e So we now have S = 1,4, ...

%e Add the smallest prime not added so far in order to get a composite:

%e - Can we add 2 (smallest available prime)? Yes, because 4+2=6 and 6 is composite.

%e So we now have S = 1,4,6, ...

%e - Can we add 5? No, because 6+5=11 and 11 is prime.

%e - Can we add 7? No, because 6+7=13 and 13 is prime.

%e - Can we add 11? No, because 6+11=17 and 17 is prime.

%e - Can we add 13? No, because 6+13=19 and 19 is prime.

%e - Can we add 17? No, because 6+17=23 and 23 is prime.

%e - Can we add 19? Yes, because 6+19=25 and 25 is composite.

%e So we now have S = 1,4,6,25, ...

%e - Can we add 5 (smallest available prime)? Yes, because 25+5=30 and 30 is composite.

%e So we now have S = 1,4,6,25,30, ...

%e etc.

%K base,easy,nonn

%O 1,2

%A _Eric Angelini_, Sep 26 2006

%E Terms computed by _Max Alekseyev_