A333365 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.
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
Offset: 0
Examples
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 ; ...
Links
- Alois P. Heinz, Rows n = 0..1000, flattened
Crossrefs
Programs
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Maple
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); -
Mathematica
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 *)
Formula
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