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A258037 Numbers k such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

%I A258037 #10 Nov 14 2024 08:20:11
%S A258037 1,2,3,5,7,9,11,13,15,16,18,20,22,24,26,27,29,31,33,35,37,39,40,42,44,
%T A258037 46,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,
%U A258037 93,95,97,100,102,104,106,108,110,112,113,115,117,119,121
%N A258037 Numbers k such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.
%C A258037 Partition of the positive integers: A258036, A258037;
%C A258037 Corresponding partition of the primes: A258038, A258039.
%C A258037 Conjecture: all the terms of the difference sequence of A258037 belong to {1,2,3}.
%H A258037 Clark Kimberling, <a href="/A258037/b258037.txt">Table of n, a(n) for n = 1..1000</a>
%F A258037 D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by _Jason Yuen_, Nov 13 2024]
%e A258037 D(prime(2), 1) = 3 - 2 > 0, so a(1) = 1;
%e A258037 D(prime(3), 2) = 5 - 2*3 + 2 > 0, so a(2) = 2;
%e A258037 D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0;
%t A258037 u = Table[Prime[Range[k]], {k, 1, 1000}];
%t A258037 v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
%t A258037 w1 = Flatten[Position[v, -1]] (* A258036 *)
%t A258037 w2 = Flatten[Position[v, 1]]  (* A258037 *)
%t A258037 p1 = Prime[w1]  (* A258038 *)
%t A258037 p2 = Prime[w2]  (* A258039 *)
%o A258037 (PARI) is(k) = {my(p=primes(k));sum(i=0,k-1,(-1)^i*p[k-i]*binomial(k-1,i))>0} \\ _Jason Yuen_, Nov 13 2024
%Y A258037 Cf. A258036, A258038, A258039.
%K A258037 nonn,easy
%O A258037 1,2
%A A258037 _Clark Kimberling_, Jun 05 2015