%I #65 Mar 02 2019 04:22:18
%S 1,0,2,-2,-4,4,10,-10,8,-8,6,16,-12,-16,12,-14,-20,28,-28,-6,22,14,20,
%T 26,-26,-22,-24,24,-18,18,32,-32,-30,40,-40,30,-38,-34,46,38,50,-46,
%U 34,68,-64,-44,58,-58,-48,-50,48,52,-54,44,-52,36,-42,56
%N Distinct second differences in the sequence of primes in order of appearance.
%C A036263 excluding repeated terms. - _Iain Fox_, Nov 30 2017
%H Robert G. Wilson v, <a href="/A295746/b295746.txt">Table of n, a(n) for n = 1..358</a> (first 97 terms from Edward Bernstein)
%e From _Jon E. Schoenfield_, Jan 15 2018: (Start)
%e The first several primes and their 1st and 2nd differences are as follows:
%e .
%e k prime(k) 1st difference 2nd difference
%e -- -------- -------------- -------------------
%e 1 2
%e 3 - 2 = 1
%e 2 3 2 - 1 = 1 (new)
%e 5 - 3 = 2
%e 3 5 2 - 2 = 0 (new)
%e 7 - 5 = 2
%e 4 7 4 - 2 = 2 (new)
%e 11 - 7 = 4
%e 5 11 2 - 4 = -2 (new)
%e 13 - 11 = 2
%e 6 13 4 - 2 = 2 (repeat)
%e 17 - 13 = 4
%e 7 17 2 - 4 = -2 (repeat)
%e 19 - 17 = 2
%e 8 19 4 - 2 = 2 (repeat)
%e 23 - 19 = 4
%e 9 23 6 - 4 = 2 (repeat)
%e 29 - 23 = 6
%e 10 29 2 - 6 = -4 (new)
%e 31 - 29 = 2
%e 11 31
%e .
%e and the 2nd differences that are not repeats of 2nd differences encountered earlier are, in order of appearance, 1, 0, 2, -2, -4, ..., i.e., the terms of this sequence. (End)
%p P:= select(isprime, [2,seq(i,i=3..10^6,2)]):
%p DP:= P[2..-1]-P[1..-2]:
%p DDP:= DP[2..-1]-DP[1..-2]:
%p ListTools:-MakeUnique(DDP); # _Robert Israel_, Jan 15 2018
%t DeleteDuplicates@ Differences[Prime@ Range[10^4], 2] (* _Michael De Vlieger_, Dec 09 2017 *)
%o (PARI) first(n) = { my(res = vector(n), i=3, j=3); res[1]=1; res[2]=0; while(i<=n, my(d=prime(j+2)+prime(j)-2*prime(j+1)); if(!setsearch(Set(res), d), res[i]=d; i++); j++); res; } \\ _Iain Fox_, Nov 30 2017
%Y Cf. A036263. A295973 are the primes associated with the new second differences.
%K sign
%O 1,3
%A _Edward Bernstein_, Nov 29 2017