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A258038 Numbers prime(k) such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

%I A258038 #8 Nov 14 2024 08:20:07
%S A258038 7,13,19,29,37,43,59,67,73,83,97,107,113,131,139,151,163,179,191,197,
%T A258038 211,223,229,239,251,263,271,281,293,311,317,337,349,359,373,383,397,
%U A258038 409,421,433,443,457,463,479,491,503,521,523,547,563,571,587,599,607
%N A258038 Numbers prime(k) such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.
%C A258038 Partition of the positive integers: A258036, A258037;
%C A258038 Corresponding partition of the primes: A258038, A258039.
%H A258038 Clark Kimberling, <a href="/A258038/b258038.txt">Table of n, a(n) for n = 1..1000</a>
%F A258038 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]
%F A258038 a(n) = prime(A258036(n)). - _Jason Yuen_, Nov 13 2024
%e A258038 D(prime(2), 1) = 3 - 2 > 0;
%e A258038 D(prime(3), 2) = 5 - 2*3 + 2 > 0;
%e A258038 D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0, so a(1) = prime(4) = 7;
%t A258038 u = Table[Prime[Range[k]], {k, 1, 1000}];
%t A258038 v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
%t A258038 w1 = Flatten[Position[v, -1]] (* A258036 *)
%t A258038 w2 = Flatten[Position[v, 1]]  (* A258037 *)
%t A258038 p1 = Prime[w1]  (* A258038 *)
%t A258038 p2 = Prime[w2]  (* A258039 *)
%Y A258038 Cf. A258036, A258037, A258039.
%K A258038 nonn,easy
%O A258038 1,1
%A A258038 _Clark Kimberling_, Jun 05 2015