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A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1). 18
1, 0, 1, 0, 1, 3, 0, 0, 1, 3, 0, 0, 1, 1, 4, 0, 1, 0, 1, 2, 4, 0, 0, 1, 1, 2, 2, 7, 0, 0, 0, 1, 1, 2, 2, 7, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 0, 1, 1, 2, 1, 4, 4, 10, 0, 0, 0, 1, 1, 1, 1, 4, 2, 4, 5, 10, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,6

LINKS

Alois P. Heinz, Antidiagonals n = 2..142, flattened

FORMULA

For k=p prime: T(n,p) = [n/p] + [n/p^2] + [n/p^3] + .... For k = p^m a prime power: T(n,p^m) = [T(n,p)/m]. For k = b*c with b and c coprime: T(n,a*b) = min(T(n,a), T(n,b)). T(n,k) is close to, but below, n/A090624(k).

EXAMPLE

Square array starts:

1, 0, 0, 0, 0, 0, 0, ...

1, 1, 0, 0, 1, 0, 0, ...

3, 1, 1, 0, 1, 0, 1, ...

3, 1, 1, 1, 1, 0, 1, ...

4, 2, 2, 1, 2, 0, 1, ...

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

7, 2, 3, 1, 2, 1, 2, ...

MAPLE

f:= proc(n, p) local c, k; c, k:= 0, p;

       while n>=k do c:= c+iquo(n, k); k:= k*p od; c

    end:

T:= (n, k)-> min(seq(iquo(f(n, i[1]), i[2]), i=ifactors(k)[2])):

seq(seq(T(n, 2+d-n), n=2..d), d=2..20);  # Alois P. Heinz, Oct 04 2012

MATHEMATICA

f[n_, p_] := Module[{c = 0, k = p}, While[n >= k , c = c + Quotient[n, k]; k = k*p ]; c ]; t[n_, k_] := Min[ Table[ Quotient[f[n, i[[1]]], i[[2]]], {i, FactorInteger[k]}]]; Table[ Table[t[n, 2 + d - n], {n, 2, d}], {d, 2, 20}] // Flatten (* Jean-François Alcover, Oct 03 2013, translated from Alois P. Heinz's Maple program *)

CROSSREFS

Columns include A011371, A054861, A090616, A027868, A054861, A054896, A090617, A090618, A027868, A064458, A090619, A090620, A054896, A027868, A090621.

Cf. A090623, A090624, A115627.

Diagonal gives A011776. - Alois P. Heinz, Oct 04 2012

Sequence in context: A163541 A280815 A165974 * A035696 A170840 A300725

Adjacent sequences:  A090619 A090620 A090621 * A090623 A090624 A090625

KEYWORD

nonn,tabl

AUTHOR

Henry Bottomley, Dec 06 2003

STATUS

approved

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Last modified September 20 03:32 EDT 2019. Contains 327209 sequences. (Running on oeis4.)