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.

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

Links

Table of n, a(n) for n=0..90.

Wikipedia, Fundamental theorem of arithmetic.

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

Adjacent sequences: A372714 A372715 A372716 * A372718 A372719 A372720

Keyword

nonn,tabl

Author

Peter Luschny, May 11 2024

Status

approved