This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376682 #17 Oct 18 2024 11:42:25
%S A376682 1,2,1,3,1,0,5,2,1,1,7,2,0,-1,-2,11,4,2,2,3,5,13,2,-2,-4,-6,-9,-14,17,
%T A376682 4,2,4,8,14,23,37,19,2,-2,-4,-8,-16,-30,-53,-90,23,4,2,4,8,16,32,62,
%U A376682 115,205,29,6,2,0,-4,-12,-28,-60,-122,-237,-442,31,2,-4,-6,-6,-2,10,38,98,220,457,899
%N A376682 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the noncomposite numbers (A008578).
%C A376682 Row k is the k-th differences of the noncomposite numbers.
%F A376682 A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A008578(i+k).
%e A376682 Array begins:
%e A376682 n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
%e A376682 -----------------------------------------------------------
%e A376682 k=0: 1 2 3 5 7 11 13 17 19
%e A376682 k=1: 1 1 2 2 4 2 4 2 4
%e A376682 k=2: 0 1 0 2 -2 2 -2 2 2
%e A376682 k=3: 1 -1 2 -4 4 -4 4 0 -6
%e A376682 k=4: -2 3 -6 8 -8 8 -4 -6 14
%e A376682 k=5: 5 -9 14 -16 16 -12 -2 20 -28
%e A376682 k=6: -14 23 -30 32 -28 10 22 -48 48
%e A376682 k=7: 37 -53 62 -60 38 12 -70 96 -70
%e A376682 k=8: -90 115 -122 98 -26 -82 166 -166 86
%e A376682 k=9: 205 -237 220 -124 -56 248 -332 252 -86
%e A376682 Triangle begins:
%e A376682 1
%e A376682 2 1
%e A376682 3 1 0
%e A376682 5 2 1 1
%e A376682 7 2 0 -1 -2
%e A376682 11 4 2 2 3 5
%e A376682 13 2 -2 -4 -6 -9 -14
%e A376682 17 4 2 4 8 14 23 37
%e A376682 19 2 -2 -4 -8 -16 -30 -53 -90
%e A376682 23 4 2 4 8 16 32 62 115 205
%e A376682 29 6 2 0 -4 -12 -28 -60 -122 -237 -442
%e A376682 31 2 -4 -6 -6 -2 10 38 98 220 457 899
%t A376682 nn=12;
%t A376682 t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
%t A376682 (* or *)
%t A376682 nn=12;
%t A376682 q=Table[If[n==0,1,Prime[n]],{n,0,2nn}];
%t A376682 Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,nn},{i,nn}]
%Y A376682 The version for modern primes (A000040) is A095195.
%Y A376682 Initial rows: A008578, A075526, A036263 with 0 prepended.
%Y A376682 Column n = 1 is A030016 (modern A007442).
%Y A376682 A version for partitions is A175804, cf. A053445, A281425, A320590.
%Y A376682 Antidiagonal-sums are A376683 (modern A140119), absolute A376684 (modern A376681).
%Y A376682 First position of 0 is A376855 (modern A376678).
%Y A376682 For composite instead of prime we have A377033.
%Y A376682 For squarefree instead of prime we have A377038, nonsquarefree A377046.
%Y A376682 For prime-power instead of composite we have A377051.
%Y A376682 A000040 lists the primes, differences A001223, second A036263.
%Y A376682 Cf. A002808, A064113, A065890, A084758, A173390, A233671, A258025, A258026, A333254, A376602 (zeros), A376651 (positives), A376652 (negatives), A376680, A377037.
%K A376682 sign,tabl
%O A376682 0,2
%A A376682 _Gus Wiseman_, Oct 15 2024