A343751
A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 5, 4, 0, 1, 10, 19, 8, 0, 1, 17, 69, 65, 16, 0, 1, 28, 188, 410, 211, 32, 0, 1, 41, 496, 1726, 2261, 665, 64, 0, 1, 58, 1029, 7182, 14343, 11970, 2059, 128, 0, 1, 77, 2015, 20559, 93345, 112371, 61909, 6305, 256, 0, 1, 100, 3478, 54814, 360612, 1139166, 848506, 315850, 19171, 512, 0
Offset: 0
A(1,3) = 10 = 5 + 3 + 2, sum of encoded compositions [0,0,1], [0,1,0], [1,0,0].
A(4,2) = 211 = 81 + 54 + 36 + 24 + 16, sum of encoded compositions [0,4], [1,3], [2,2], [3,1], [4,0].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 2, 5, 10, 17, 28, 41, ...
0, 4, 19, 69, 188, 496, 1029, ...
0, 8, 65, 410, 1726, 7182, 20559, ...
0, 16, 211, 2261, 14343, 93345, 360612, ...
0, 32, 665, 11970, 112371, 1139166, 5827122, ...
0, 64, 2059, 61909, 848506, 13379332, 89131918, ...
-
A:= proc(n, k) option remember; `if`(n=0, 1,
`if`(k=0, 0, add(ithprime(k)^i*A(n-i, k-1), i=0..n)))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
`if`(k=0, 0, ithprime(k)*A(n-1, k)+A(n, k-1)))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
-
A[n_, k_] := A[n, k] = If[n == 0, 1,
If[k == 0, 0, Prime[k] A[n-1, k] + A[n, k-1]]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 2nd Maple program *)
A124960
Triangle read by rows: T(n,k) = p(k)*T(n-1,k) + T(n-1,k-1) (1 <= k <= n), where p(k) denotes the k-th prime.
Original entry on oeis.org
1, 2, 1, 4, 5, 1, 8, 19, 10, 1, 16, 65, 69, 17, 1, 32, 211, 410, 188, 28, 1, 64, 665, 2261, 1726, 496, 41, 1, 128, 2059, 11970, 14343, 7182, 1029, 58, 1, 256, 6305, 61909, 112371, 93345, 20559, 2015, 77, 1, 512, 19171, 315850, 848506, 1139166, 360612, 54814, 3478, 100, 1
Offset: 1
Triangle starts:
1;
2, 1;
4, 5, 1;
8, 19, 10, 1;
16, 65, 69, 17, 1;
32, 211, 410, 188, 28, 1;
-
function T(n,k)
if k lt 1 or k gt n then return 0;
elif n eq 1 and k eq 1 then return 1;
else return NthPrime(k)*T(n-1,k) + T(n-1,k-1);
end if;
return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 19 2019
-
T:=proc(n,k): if n=1 and k=1 then 1 elif k<1 or k>n then 0 else ithprime(k)*T(n-1,k)+T(n-1,k-1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
-
T[n_, k_]:= T[n, k]= If[n==1 && k==1 , 1, If[k<1 || k>n, 0, Prime[k]*T[n-1, k] + T[n-1, k-1] ]]; Table[T[n, k], {n,12}, {k, n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)
-
T(n,k) = if(n==1 && k==1, 1, if(k<1 || k>n, 0, prime(k)*T(n-1, k) + T(n-1, k-1) )); \\ G. C. Greubel, Nov 19 2019
-
@CachedFunction
def T(n,k):
if (k<1 or k>n): return 0
elif (n==1 and k==1): return 1
else: return nth_prime(k)*T(n-1, k) + T(n-1, k-1)
[[T(n,k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 19 2019
A330394
Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.
Original entry on oeis.org
1, 2, 4, 6, 9, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 2401, 32, 48, 72, 80, 108, 112, 120, 162
Offset: 0
Triangle T(n,k) begins:
1;
2;
4, 6, 9;
8, 12, 18, 20, 27, 30, 45, 50, 75, 125;
...
Last elements of rows give
A307539.
-
b:= proc(n, i) option remember; `if`(n=0, [1], [seq(
map(x-> x*ithprime(j), b(n-1, j))[], j=1..i)])
end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..5); # Alois P. Heinz, Mar 03 2020
-
t = Table[Union[Apply[Times, Tuples[Prime[Range[n]], {n}], {1}]], {n, 0, 5}];
t // TableForm
Flatten[t]
Showing 1-3 of 3 results.
Comments