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
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 >= 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
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_] := 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 ++);); x;} \\ Michel Marcus, Oct 28 2018
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