A319581 Square array T(n, k) = 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.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0

Offset

1,61

Comments

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

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

Links

Table of n, a(n) for n=1..90.

Formula

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

T(n, n) = omega(n) = A001221(n) = the number of distinct primes dividing n.

a(n) = log_2(A319582(n)).

Example

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 >= 1 > 0] + [1 >= 0 > 0] + [1 >= 2 > 0]
  = 1 + 0 + 0
  = 1.
Array begins (zeros replaced by dots):
     k =                 1 1 1
   n   1 2 3 4 5 6 7 8 9 0 1 2
   =  ------------------------
   1 | . . . . . . . . . . . .
   2 | . 1 . . . 1 . . . 1 . .
   3 | . . 1 . . 1 . . . . . 1
   4 | . 1 . 1 . 1 . . . 1 . 1
   5 | . . . . 1 . . . . 1 . .
   6 | . 1 1 . . 2 . . . 1 . 1
   7 | . . . . . . 1 . . . . .
   8 | . 1 . 1 . 1 . 1 . 1 . 1
   9 | . . 1 . . 1 . . 1 . . 1
  10 | . 1 . . 1 1 . . . 2 . .
  11 | . . . . . . . . . . 1 .
  12 | . 1 1 1 . 2 . . . 1 . 2

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_] := 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}]

Prog

(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 ++); ); x; } \\ Michel Marcus, Oct 28 2018

Crossrefs

Cf. A319582 (a multiplicative variant).

Cf. A001221.

Sequence in context: A225783 A135468 A003196 * A331302 A062977 A357879

Adjacent sequences: A319578 A319579 A319580 * A319582 A319583 A319584

Keyword

nonn,tabl

Author

Luc Rousseau, Sep 23 2018

Status

approved