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 A377051 #8 Oct 22 2024 08:00:16
%S A377051 1,2,1,3,1,0,4,1,0,0,5,1,0,0,0,7,2,1,1,1,1,8,1,-1,-2,-3,-4,-5,9,1,0,1,
%T A377051 3,6,10,15,11,2,1,1,0,-3,-9,-19,-34,13,2,0,-1,-2,-2,1,10,29,63,16,3,1,
%U A377051 1,2,4,6,5,-5,-34,-97,17,1,-2,-3,-4,-6,-10,-16,-21,-16,18,115
%N A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.
%C A377051 Row k of the array is the k-th differences of A000961.
%F A377051 A(i,j) = Sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*A000961(i+k).
%e A377051 Array form:
%e A377051 n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
%e A377051 ----------------------------------------------------------
%e A377051 k=0: 1 2 3 4 5 7 8 9 11
%e A377051 k=1: 1 1 1 1 2 1 1 2 2
%e A377051 k=2: 0 0 0 1 -1 0 1 0 1
%e A377051 k=3: 0 0 1 -2 1 1 -1 1 -3
%e A377051 k=4: 0 1 -3 3 0 -2 2 -4 6
%e A377051 k=5: 1 -4 6 -3 -2 4 -6 10 -8
%e A377051 k=6: -5 10 -9 1 6 -10 16 -18 5
%e A377051 k=7: 15 -19 10 5 -16 26 -34 23 9
%e A377051 k=8: -34 29 -5 -21 42 -60 57 -14 -42
%e A377051 k=9: 63 -34 -16 63 -102 117 -71 -28 104
%e A377051 Triangle form:
%e A377051 1
%e A377051 2 1
%e A377051 3 1 0
%e A377051 4 1 0 0
%e A377051 5 1 0 0 0
%e A377051 7 2 1 1 1 1
%e A377051 8 1 -1 -2 -3 -4 -5
%e A377051 9 1 0 1 3 6 10 15
%e A377051 11 2 1 1 0 -3 -9 -19 -34
%e A377051 13 2 0 -1 -2 -2 1 10 29 63
%e A377051 16 3 1 1 2 4 6 5 -5 -34 -97
%t A377051 nn=12;
%t A377051 t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
%t A377051 Table[t[[j,i-j+1]],{i,nn},{j,i}]
%Y A377051 Row k=0 is A000961, exclusive A246655.
%Y A377051 Row k=1 is A057820.
%Y A377051 Row k=2 is A376596.
%Y A377051 The version for primes is A095195, noncomposites A376682, composites A377033.
%Y A377051 A version for partitions is A175804, cf. A053445, A281425, A320590.
%Y A377051 For squarefree numbers we have A377038, nonsquarefree A377046.
%Y A377051 Triangle row-sums are A377052, absolute version A377053.
%Y A377051 Column n = 1 is A377054, for primes A007442 or A030016.
%Y A377051 First position of 0 in each row is A377055.
%Y A377051 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377051 A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
%Y A377051 Cf. A025475, A053707, A093555, A174965, A361102, A376340, A376598, A376653.
%K A377051 sign,tabl
%O A377051 0,2
%A A377051 _Gus Wiseman_, Oct 20 2024