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A333365
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T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.
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6
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0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 0, 0, 1, 5, 1, 1, 6, 2, 0, 0, 0, 1, 7, 2, 0, 1, 9, 2, 1, 10, 3, 1, 12, 3, 1, 0, 0, 0, 1, 14, 3, 1, 1, 17, 4, 1, 0, 0, 0, 0, 1, 19, 5, 1, 1, 23, 5, 1, 1, 26, 6, 2, 0, 1, 30, 7, 2, 0, 0, 0, 0, 0, 1
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OFFSET
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0,12
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LINKS
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Alois P. Heinz, Rows n = 0..1000, flattened
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FORMULA
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T(n,pi(n)) = A010051(n) for n > 1.
T(p,pi(p)) = 1 if p is prime.
T(prime(k),k) = 1 for k >= 1.
Recursion: T(n,k) = Sum_{q=k..pi(n-p)} T(n-p, q) with p := prime(k) and T(n,k) = 0 if n < p, or 1 if n = p. - David James Sycamore, Mar 28 2020
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EXAMPLE
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In the A000607(11) = 6 partitions of 11 into prime parts, (11), 335, 227, 2225, 2333, 22223 the least parts are 11 = prime(5) (once), 3 = prime(2)(once), and 2 = prime(1) (four times), whereas 5 and 7 (prime(3) and prime(4)) do not occur. Thus row 11 is [4,1,0,0,1].
Triangle T(n,k) begins:
0 ;
0 ;
1 ;
0, 1 ;
1 ;
1, 0, 1 ;
1, 1 ;
2, 0, 0, 1 ;
2, 1 ;
3, 1 ;
3, 1, 1 ;
4, 1, 0, 0, 1 ;
5, 1, 1 ;
6, 2, 0, 0, 0, 1 ;
7, 2, 0, 1 ;
9, 2, 1 ;
10, 3, 1 ;
12, 3, 1, 0, 0, 0, 1 ;
14, 3, 1, 1 ;
17, 4, 1, 0, 0, 0, 0, 1 ;
19, 5, 1, 1 ;
...
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MAPLE
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b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
end:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i),
coeff(p, x, i), [][]), i=2..max(2, degree(p))))(b(n, 2, x))
end:
seq(T(n), n=0..23);
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MATHEMATICA
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b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, With[{q = NextPrime[p]}, Sum[b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]]]];
T[n_] := If[n < 2, {0}, MapIndexed[If[PrimeQ[#2[[1]]], #1, Nothing]&, Rest @ CoefficientList[b[n, 2, x], x]]];
T /@ Range[0, 23] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)
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CROSSREFS
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Columns k=1-2 give: A000607(n-2) for n>1, A099773(n-3) for n>2.
Row sums give A000607 for n>0.
Length of n-th row is A000720(A331634(n)) for n>1.
Indices of rows without 1's: A330433.
Cf. A000040, A000720, A010051, A333129, A333238, A333259.
Sequence in context: A292598 A079113 A144874 * A303065 A325406 A257900
Adjacent sequences: A333362 A333363 A333364 * A333366 A333367 A333368
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KEYWORD
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nonn,tabf
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AUTHOR
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Alois P. Heinz, Mar 16 2020
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STATUS
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approved
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