login
A372717
Triangle read by rows: If k is prime then T(n, k) is the exponent of the highest power of k that divides n. T(0, 0) = T(1, 1) = 1. In all other cases T(n, k) = 0.
0
1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
OFFSET
0,13
FORMULA
n = Product_{k=0..n} k^T(n, k). (Fundamental theorem of arithmetic.)
EXAMPLE
Triangle begins:
[ 0] 1;
[ 1] 0, 1;
[ 2] 0, 0, 1;
[ 3] 0, 0, 0, 1;
[ 4] 0, 0, 2, 0, 0;
[ 5] 0, 0, 0, 0, 0, 1;
[ 6] 0, 0, 1, 1, 0, 0, 0;
[ 7] 0, 0, 0, 0, 0, 0, 0, 1;
[ 8] 0, 0, 3, 0, 0, 0, 0, 0, 0;
[ 9] 0, 0, 0, 2, 0, 0, 0, 0, 0, 0;
[10] 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0;
MAPLE
Ord := proc(n, k)
if n = 0 then return 1 fi;
if n = 1 then return k fi;
if isprime(k) then padic:-ordp(n, k) else 0 fi end:
seq(seq(Ord(n, k), k = 0..n), n = 0..12);
MATHEMATICA
{{1}, {0, 1}}~Join~Table[If[PrimeQ[k], IntegerExponent[n, k], 0], {n, 2, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 11 2024 *)
PROG
(SageMath)
def T(n, k):
if n == 0: return 1
if n == 1: return k
return 0 if not is_prime(k) else n.valuation(k)
for n in srange(11): print([T(n, k) for k in range(n+1)])
CROSSREFS
Cf. A001477.
Cf. A001222 (row sums for n>=2), A001414(n) = Sum_{k=0..n} k*T(n, k) (for n>=2).
T(n, n) = A010051(n) (prime indicator for n>=2), T(2*n, n) = T(n, n) (for n>=3).
Cf. A007814 (ruler seq. for n>=2), A007949 (3-adic valuation).
Sequence in context: A351439 A102354 A370222 * A349615 A157228 A193138
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 11 2024
STATUS
approved