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A258036 Numbers k such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

%I A258036 #10 Nov 14 2024 08:19:58
%S A258036 4,6,8,10,12,14,17,19,21,23,25,28,30,32,34,36,38,41,43,45,47,48,50,52,
%T A258036 54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,
%U A258036 99,101,103,105,107,109,111,114,116,118,120,122,124,126,128
%N A258036 Numbers k such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.
%C A258036 Partition of the positive integers: A258036, A258037;
%C A258036 Corresponding partition of the primes: A258038, A258039.
%C A258036 Do all the terms of the difference sequence of A258036 belong to {1,2,3}?
%H A258036 Clark Kimberling, <a href="/A258036/b258036.txt">Table of n, a(n) for n = 1..1000</a>
%F A258036 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 A258036 D(prime(2), 1) = 3 - 2 > 0;
%e A258036 D(prime(3), 2) = 5 - 2*3 + 2 > 0;
%e A258036 D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0, so a(1) = 4;
%t A258036 u = Table[Prime[Range[k]], {k, 1, 1000}];
%t A258036 v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
%t A258036 w1 = Flatten[Position[v, -1]] (* A258036 *)
%t A258036 w2 = Flatten[Position[v, 1]]  (* A258037 *)
%t A258036 Prime[w1]  (* A258038 *)
%t A258036 Prime[w2]  (* A258039 *)
%o A258036 (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 A258036 Cf. A258037, A258038, A258039.
%K A258036 nonn,easy
%O A258036 1,1
%A A258036 _Clark Kimberling_, Jun 05 2015