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First position of 0 in each row is A377037.
A000040 lists the primes, differences A001223, second seconds A036263.
A008578 lists the noncomposites, differences A075526, seconds A036263 with 0 prepended.
Cf. `A054819, A065310, A065890, `A084758, A173390, `A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.
allocated for Gus Wiseman
Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).
4, 6, 2, 8, 2, 0, 9, 1, -1, -1, 10, 1, 0, 1, 2, 12, 2, 1, 1, 0, -2, 14, 2, 0, -1, -2, -2, 0, 15, 1, -1, -1, 0, 2, 4, 4, 16, 1, 0, 1, 2, 2, 0, -4, -8, 18, 2, 1, 1, 0, -2, -4, -4, 0, 8, 20, 2, 0, -1, -2, -2, 0, 4, 8, 8, 0, 21, 1, -1, -1, 0, 2, 4, 4, 0, -8, -16, -16
0,1
Row n is the k-th differences of A002808 = the composite numbers.
A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A002808(i+k).
Array begins:
n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
----------------------------------------------------------
k=0: 4 6 8 9 10 12 14 15 16
k=1: 2 2 1 1 2 2 1 1 2
k=2: 0 -1 0 1 0 -1 0 1 0
k=3: -1 1 1 -1 -1 1 1 -1 -1
k=4: 2 0 -2 0 2 0 -2 0 2
k=5: -2 -2 2 2 -2 -2 2 2 -2
k=6: 0 4 0 -4 0 4 0 -4 -1
k=7: 4 -4 -4 4 4 -4 -4 3 10
k=8: -8 0 8 0 -8 0 7 7 -29
k=9: 8 8 -8 -8 8 7 0 -36 63
Triangle begins:
4
6 2
8 2 0
9 1 -1 -1
10 1 0 1 2
12 2 1 1 0 -2
14 2 0 -1 -2 -2 0
15 1 -1 -1 0 2 4 4
16 1 0 1 2 2 0 -4 -8
18 2 1 1 0 -2 -4 -4 0 8
20 2 0 -1 -2 -2 0 4 8 8 0
21 1 -1 -1 0 2 4 4 0 -8 -16 -16
nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1, PrimeQ]&, 4, 2*nn], k], nn], {k, 0, nn}]
Initial rows: A002808, A073783, A073445.
The version for primes is A095195 or A376682.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377034, absolute version A377035.
Column n = 1 is A377036, for primes A007442 or A030016.
First position of 0 is A377037.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A008578 lists the noncomposites, differences A075526, seconds A036263 with 0 prepended.
Cf. A018252, `A064113, `A065890, `A084758, `A173390, `A333214, `A333215, `A333254, `A350004, A376680.
Cf: `A054819, A065310, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative).
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Gus Wiseman, Oct 17 2024
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allocated for Gus Wiseman
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