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A090624
If n = Product(pj^ej), i.e., written in its prime factorization, then a(n) = max_j{(pj-1)*ej}.
6
1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 2, 12, 6, 4, 4, 16, 4, 18, 4, 6, 10, 22, 3, 8, 12, 6, 6, 28, 4, 30, 5, 10, 16, 6, 4, 36, 18, 12, 4, 40, 6, 42, 10, 4, 22, 46, 4, 12, 8, 16, 12, 52, 6, 10, 6, 18, 28, 58, 4, 60, 30, 6, 6, 12, 10, 66, 16, 22, 6, 70, 4, 72, 36, 8, 18, 10, 12, 78, 4, 8, 40, 82, 6
OFFSET
2,2
COMMENTS
The highest power of k dividing n! (A090622) is close to, but below, n/a(k).
LINKS
A. M. Oller-Marcen and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011) 11.6.8, propos. 3.
FORMULA
a(p) = p-1; a(p^m) = (p-1)*m.
a(b*c) = max(a(b), a(c)) for b and c coprime.
a(n) = lim_{k->inf} k/A090622(k, n) = lim_{k->inf} (k/highest power of k dividing n!). - David W. Wilson, Sep 05 2016
EXAMPLE
72 = 2^3*3^2 so a(72) = max((2-1)*3, (3-1)*2) = max(3,4) = 4.
MAPLE
seq(max(map(t ->(t[1]-1)*t[2], ifactors(n)[2])), n=2..100); # Robert Israel, Sep 06 2016
MATHEMATICA
a[n_] := Module[{aux = FactorInteger[n]}, Max[Table[aux[[i, 2]]*(aux[[i, 1]] - 1), {i, 1, Length[aux]}]]] (* José María Grau Ribas, Feb 15 2010 *)
PROG
(PARI) a(n)=my(f=factor(n)); vecmax(vector(#f~, i, (f[i, 1]-1)*f[i, 2])) \\ Charles R Greathouse IV, Sep 07 2016
(Python)
from sympy import factorint
def A090624(n): return max((p-1)*e for p, e in factorint(n).items()) # Chai Wah Wu, Apr 28 2023
CROSSREFS
Sequence in context: A286472 A279690 A167272 * A304407 A099735 A322352
KEYWORD
nonn
AUTHOR
Henry Bottomley, Dec 06 2003
STATUS
approved