A319582 Square array T(n, k) = 2 ^ (Sum_{p prime} [v_p(n) >= v_p(k) > 0]) read by antidiagonals up, where [] is the Iverson bracket and v_p is the p-adic valuation, n >= 1, k >= 1.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1

Offset: 1

Author

Luc Rousseau, Sep 24 2018

Keywords

• nonn
• tabl

Comments

T(n, k) is the number of integers located on the sphere with center n and radius log(k), when the metric is given by log(A089913).

T(., k) is multiplicative and k-periodic.

T(n, .) is multiplicative and n^2-periodic.

Examples

			T(60, 50) = T(2^2 * 3^1 * 5^1, 2^1 * 5^2)
  = T(2^2, 2^1) * T(3^1, 3^0) * T(5^1, 5^2)
  = 2^[2 >= 1 > 0] * 2^[1 >= 0 > 0] * 2^[1 >= 2 > 0]
  = 2^1 * 2^0 * 2^0 = 2 * 1 * 1 = 2.
Array begins:
     k =                 1 1 1
   n   1 2 3 4 5 6 7 8 9 0 1 2
   =  ------------------------
   1 | 1 1 1 1 1 1 1 1 1 1 1 1
   2 | 1 2 1 1 1 2 1 1 1 2 1 1
   3 | 1 1 2 1 1 2 1 1 1 1 1 2
   4 | 1 2 1 2 1 2 1 1 1 2 1 2
   5 | 1 1 1 1 2 1 1 1 1 2 1 1
   6 | 1 2 2 1 1 4 1 1 1 2 1 2
   7 | 1 1 1 1 1 1 2 1 1 1 1 1
   8 | 1 2 1 2 1 2 1 2 1 2 1 2
   9 | 1 1 2 1 1 2 1 1 2 1 1 2
  10 | 1 2 1 1 2 2 1 1 1 4 1 1
  11 | 1 1 1 1 1 1 1 1 1 1 2 1
  12 | 1 2 2 2 1 4 1 1 1 2 1 4
		

Crossrefs

Cf. A319581 (an additive variant).

Cf. A001221, A034444, A089913.

Programs

Mathematica

F[n_] := If[n == 1, {}, FactorInteger[n]]
V[p_] := If[KeyExistsQ[#, p], #[p], 0] &
PreT[n_, k_] :=
Module[{fn = F[n], fk = F[k], p, an = <||>, ak = <||>, w},
  p = Union[First /@ fn, First /@ fk];
  (an[#[[1]]] = #[[2]]) & /@ fn;
  (ak[#[[1]]] = #[[2]]) & /@ fk;
  w = ({V[#][an], V[#][ak]}) & /@ p;
  Select[w, (#[[1]] >= #[[2]] > 0) &]
  ]
T[n_, k_] := 2^Length[PreT[n, k]]
A004736[n_] := Binomial[Floor[3/2 + Sqrt[2*n]], 2] - n + 1
A002260[n_] := n - Binomial[Floor[1/2 + Sqrt[2*n]], 2]
a[n_] := T[A004736[n], A002260[n]]
Table[a[n], {n, 1, 90}]

PARI

maxp(n) = if (n==1, 1, vecmax(factor(n)[,1]));
T(n, k) = {pmax = max(maxp(n), maxp(k)); x = 0; forprime(p=2, pmax, if ((valuation(n, p) >= valuation(k, p)) && (valuation(k, p) > 0), x ++);); 2^x;} \\ Michel Marcus, Oct 28 2018

Formula

T(n, k) = card({x; d(n, x) = log(k)}), if d denotes log(A089913(., .)), which is a metric.

T(n, k) = 2 ^ (Sum_{p prime} [v_p(n) >= v_p(k) > 0]).

a(n) = 2 ^ A319581(n).

T(n, n) = 2 ^ A001221(n) = A034444(n).