A319582 - OEIS

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

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OFFSET

1,5

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.

LINKS

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

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).

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^[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

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}]

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 ++); ); 2^x; } \\ Michel Marcus, Oct 28 2018

CROSSREFS

Cf. A319581 (an additive variant).

Cf. A001221, A034444, A089913.

Sequence in context: A060209 A037830 A174353 * A187447 A146292 A139039

Adjacent sequences: A319579 A319580 A319581 * A319583 A319584 A319585

KEYWORD

nonn,tabl

AUTHOR

Luc Rousseau, Sep 24 2018

STATUS

approved