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%I A377046 #7 Oct 19 2024 21:43:16
%S A377046 4,8,4,9,1,-3,12,3,2,5,16,4,1,-1,-6,18,2,-2,-3,-2,4,20,2,0,2,5,7,3,24,
%T A377046 4,2,2,0,-5,-12,-15,25,1,-3,-5,-7,-7,-2,10,25,27,2,1,4,9,16,23,25,15,
%U A377046 -10,28,1,-1,-2,-6,-15,-31,-54,-79,-94,-84,32,4,3,4,6,12,27,58,112,191,285,369
%N A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.
%C A377046 Row k is the k-th differences of A013929.
%F A377046 A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A013929(i+k).
%e A377046 Array form:
%e A377046         n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
%e A377046   ---------------------------------------------------------
%e A377046   k=0:   4     8     9    12    16    18    20    24    25
%e A377046   k=1:   4     1     3     4     2     2     4     1     2
%e A377046   k=2:  -3     2     1    -2     0     2    -3     1    -1
%e A377046   k=3:   5    -1    -3     2     2    -5     4    -2     4
%e A377046   k=4:  -6    -2     5     0    -7     9    -6     6    -7
%e A377046   k=5:   4     7    -5    -7    16   -15    12   -13    10
%e A377046   k=6:   3   -12    -2    23   -31    27   -25    23   -13
%e A377046   k=7: -15    10    25   -54    58   -52    48   -36    13
%e A377046   k=8:  25    15   -79   112  -110   100   -84    49     1
%e A377046   k=9: -10   -94   191  -222   210  -184   133   -48   -57
%e A377046 Triangle form:
%e A377046    4
%e A377046    8   4
%e A377046    9   1  -3
%e A377046   12   3   2   5
%e A377046   16   4   1  -1  -6
%e A377046   18   2  -2  -3  -2   4
%e A377046   20   2   0   2   5   7   3
%e A377046   24   4   2   2   0  -5 -12 -15
%e A377046   25   1  -3  -5  -7  -7  -2  10  25
%e A377046   27   2   1   4   9  16  23  25  15 -10
%e A377046   28   1  -1  -2  -6 -15 -31 -54 -79 -94 -84
%e A377046   32   4   3   4   6  12  27  58 112 191 285 369
%t A377046 nn=9;
%t A377046 t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}]
%t A377046 Table[t[[j,i-j+1]],{i,nn},{j,i}]
%Y A377046 Initial rows: A013929, A078147, A376593.
%Y A377046 The version for primes is A095195, noncomposites A376682, composites A377033.
%Y A377046 A version for partitions is A175804, cf. A053445, A281425, A320590.
%Y A377046 For squarefree numbers we have A377038, sums A377039, absolute A377040.
%Y A377046 Triangle row-sums are A377047, absolute version A377048.
%Y A377046 Column n = 1 is A377049, for squarefree A377041, for prime A007442 or A030016.
%Y A377046 First position of 0 in each row is A377050.
%Y A377046 For prime-power instead of nonsquarefree we have A377051.
%Y A377046 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377046 A005117 lists the squarefree numbers.
%Y A377046 A073576 counts integer partitions into squarefree numbers, factorizations A050320.
%Y A377046 Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A120992, A376311, A376591, A376592.
%K A377046 sign,tabl
%O A377046 0,1
%A A377046 _Gus Wiseman_, Oct 19 2024

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