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Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.
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%I #8 Oct 22 2024 08:00:16

%S 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 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 1,2,4,6,5,-5,-34,-97,17,1,-2,-3,-4,-6,-10,-16,-21,-16,18,115

%N 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 Row k of the array is the k-th differences of A000961.

%F A(i,j) = Sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*A000961(i+k).

%e Array form:

%e n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:

%e ----------------------------------------------------------

%e k=0: 1 2 3 4 5 7 8 9 11

%e k=1: 1 1 1 1 2 1 1 2 2

%e k=2: 0 0 0 1 -1 0 1 0 1

%e k=3: 0 0 1 -2 1 1 -1 1 -3

%e k=4: 0 1 -3 3 0 -2 2 -4 6

%e k=5: 1 -4 6 -3 -2 4 -6 10 -8

%e k=6: -5 10 -9 1 6 -10 16 -18 5

%e k=7: 15 -19 10 5 -16 26 -34 23 9

%e k=8: -34 29 -5 -21 42 -60 57 -14 -42

%e k=9: 63 -34 -16 63 -102 117 -71 -28 104

%e Triangle form:

%e 1

%e 2 1

%e 3 1 0

%e 4 1 0 0

%e 5 1 0 0 0

%e 7 2 1 1 1 1

%e 8 1 -1 -2 -3 -4 -5

%e 9 1 0 1 3 6 10 15

%e 11 2 1 1 0 -3 -9 -19 -34

%e 13 2 0 -1 -2 -2 1 10 29 63

%e 16 3 1 1 2 4 6 5 -5 -34 -97

%t nn=12;

%t t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]

%t Table[t[[j,i-j+1]],{i,nn},{j,i}]

%Y Row k=0 is A000961, exclusive A246655.

%Y Row k=1 is A057820.

%Y Row k=2 is A376596.

%Y The version for primes is A095195, noncomposites A376682, composites A377033.

%Y A version for partitions is A175804, cf. A053445, A281425, A320590.

%Y For squarefree numbers we have A377038, nonsquarefree A377046.

%Y Triangle row-sums are A377052, absolute version A377053.

%Y Column n = 1 is A377054, for primes A007442 or A030016.

%Y First position of 0 in each row is A377055.

%Y A000040 lists the primes, differences A001223, seconds A036263.

%Y A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.

%Y Cf. A025475, A053707, A093555, A174965, A361102, A376340, A376598, A376653.

%K sign,tabl

%O 0,2

%A _Gus Wiseman_, Oct 20 2024