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A257232
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Triangle T(n, k) with the natural numbers in columns with nonprime k and the nonnegative numbers in columns with prime k, for 1 <= k <= n.
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3
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1, 2, 0, 3, 1, 0, 4, 2, 1, 1, 5, 3, 2, 2, 0, 6, 4, 3, 3, 1, 1, 7, 5, 4, 4, 2, 2, 0, 8, 6, 5, 5, 3, 3, 1, 1, 9, 7, 6, 6, 4, 4, 2, 2, 1, 10, 8, 7, 7, 5, 5, 3, 3, 2, 1, 11, 9, 8, 8, 6, 6, 4, 4, 3, 2, 0
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OFFSET
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1,2
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COMMENTS
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Row n ends in a 0 if n is prime; otherwise it ends in 1.
The alternating row sums give 1, seven times 2, six times 3, six times 4, four times 5, twice 6, ..., and the multiplicity sequence 1, 7, 6, 6, 4, 2, ... is given in A257233.
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LINKS
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FORMULA
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T(n, k) = n - (k-1) - [isprime(k)], with [isprime(k)] = A010051(k), for 1 <= k <= n.
O.g.f. for column k (with leading zeros): x^k/(1-x)^2 if k is nonprime, otherwise x^(k+1)/(1-x)^2.
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EXAMPLE
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The triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 11 ...
1: 1
2: 2 0
3: 3 1 0
4: 4 2 1 1
5: 5 3 2 2 0
6: 6 4 3 3 1 1
7: 7 5 4 4 2 2 0
8: 8 6 5 5 3 3 1 1
9: 9 7 6 6 4 4 2 2 1
10: 10 8 7 7 5 5 3 3 2 1
11: 11 9 8 8 6 6 4 4 3 2 0
...
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MATHEMATICA
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Table[n - (k - 1) - Boole[PrimeQ@ k], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Apr 19 2015 *)
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PROG
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(Haskell)
a257232 n k = a257232_tabl !! (n-1) !! (k-1)
a257232_row n = a257232_tabl !! (n-1)
a257232_tabl = iterate
(\xs@(x:_) -> map (+ 1) xs ++ [1 - a010051 (x + 1)]) [1]
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CROSSREFS
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Cf. A256885 (row sums), A257233 (multiplicities for alternating row sums).
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KEYWORD
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AUTHOR
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STATUS
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approved
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