A319582 - OEIS

%I #20 Dec 14 2024 09:15:16

%S 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,

%T 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,

%U 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,2,1,1

%N 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.

%C 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).

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

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

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

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

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

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

%e T(60, 50) = T(2^2 * 3^1 * 5^1, 2^1 * 5^2)

%e   = T(2^2, 2^1) * T(3^1, 3^0) * T(5^1, 5^2)

%e   = 2^[2 >= 1 > 0] * 2^[1 >= 0 > 0] * 2^[1 >= 2 > 0]

%e   = 2^1 * 2^0 * 2^0 = 2 * 1 * 1 = 2.

%e Array begins:

%e      k =                 1 1 1

%e    n   1 2 3 4 5 6 7 8 9 0 1 2

%e    =  ------------------------

%e    1 | 1 1 1 1 1 1 1 1 1 1 1 1

%e    2 | 1 2 1 1 1 2 1 1 1 2 1 1

%e    3 | 1 1 2 1 1 2 1 1 1 1 1 2

%e    4 | 1 2 1 2 1 2 1 1 1 2 1 2

%e    5 | 1 1 1 1 2 1 1 1 1 2 1 1

%e    6 | 1 2 2 1 1 4 1 1 1 2 1 2

%e    7 | 1 1 1 1 1 1 2 1 1 1 1 1

%e    8 | 1 2 1 2 1 2 1 2 1 2 1 2

%e    9 | 1 1 2 1 1 2 1 1 2 1 1 2

%e   10 | 1 2 1 1 2 2 1 1 1 4 1 1

%e   11 | 1 1 1 1 1 1 1 1 1 1 2 1

%e   12 | 1 2 2 2 1 4 1 1 1 2 1 4

%t F[n_] := If[n == 1, {}, FactorInteger[n]]

%t V[p_] := If[KeyExistsQ[#, p], #[p], 0] &

%t PreT[n_, k_] :=

%t Module[{fn = F[n], fk = F[k], p, an = <||>, ak = <||>, w},

%t   p = Union[First /@ fn, First /@ fk];

%t   (an[#[[1]]] = #[[2]]) & /@ fn;

%t   (ak[#[[1]]] = #[[2]]) & /@ fk;

%t   w = ({V[#][an], V[#][ak]}) & /@ p;

%t   Select[w, (#[[1]] >= #[[2]] > 0) &]

%t   ]

%t T[n_, k_] := 2^Length[PreT[n, k]]

%t A004736[n_] := Binomial[Floor[3/2 + Sqrt[2*n]], 2] - n + 1

%t A002260[n_] := n - Binomial[Floor[1/2 + Sqrt[2*n]], 2]

%t a[n_] := T[A004736[n], A002260[n]]

%t Table[a[n], {n, 1, 90}]

%o (PARI) maxp(n) = if (n==1, 1, vecmax(factor(n)[,1]));

%o 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

%Y Cf. A319581 (an additive variant).

%Y Cf. A001221, A034444, A089913.

%K nonn,tabl

%O 1,5

%A _Luc Rousseau_, Sep 24 2018