The On-Line Encyclopedia of Integer Sequences (OEIS)

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A322027

Maximum order of primeness among the prime factors of n; a(1) = 0.
(history; published version)

#11 by Alois P. Heinz at Wed Nov 28 19:04:23 EST 2018

STATUS

editing

approved

#10 by Alois P. Heinz at Wed Nov 28 19:04:20 EST 2018

LINKS

Alois P. Heinz, <a href="/A322027/b322027.txt">Table of n, a(n) for n = 1..65536</a>

STATUS

approved

editing

#9 by Alois P. Heinz at Sun Nov 25 07:47:27 EST 2018

STATUS

proposed

approved

#8 by Alois P. Heinz at Sat Nov 24 14:51:32 EST 2018

STATUS

editing

proposed

#7 by Alois P. Heinz at Sat Nov 24 14:51:22 EST 2018

NAME

Maximum order of primeness among the prime factors of n; a(1) = 0.

MAPLE

seq(a(n), n=1..120); # _Alois P. Heinz_, Nov 24 2018

#6 by Alois P. Heinz at Sat Nov 24 14:47:42 EST 2018

MAPLE

with(numtheory):

p:= proc(n) option remember;

`if`(isprime(n), 1+p(pi(n)), 0)

end:

a:= n-> max(0, map(p, factorset(n))):

seq(a(n), n=1..120);

STATUS

proposed

editing

Discussion

Sat Nov 24

14:50

Alois P. Heinz: 1 does not have prime factors, so there is no primeness, and max of the empty set is -infinity.  a(1) has to be defined separately.

#5 by Gus Wiseman at Sat Nov 24 13:01:53 EST 2018

STATUS

editing

proposed

#4 by Gus Wiseman at Sat Nov 24 13:01:37 EST 2018

LINKS

N. Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included at A006450 with permission of the author]

#3 by Gus Wiseman at Sat Nov 24 11:41:53 EST 2018

CROSSREFS

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A303431, A316470A322028, A322030.

#2 by Gus Wiseman at Sat Nov 24 11:20:27 EST 2018

NAME

allocated for Gus WisemanMaximum order of primeness among the prime factors of n.

DATA

0, 1, 2, 1, 3, 2, 1, 1, 2, 3, 4, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 4, 1, 2, 3, 1, 2, 1, 1, 3, 5, 1, 4, 2, 3, 2, 1, 1, 2, 3, 2, 2, 1, 4, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 4, 1, 2, 1, 3, 3, 1, 5, 2, 1, 3, 4, 2, 2, 2, 3, 1, 2, 1, 1, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 3, 1, 2

OFFSET

1,3

COMMENTS

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

LINKS

N. Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>

N. Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included with permission of the author]

EXAMPLE

a(105) = 3 because the prime factor of 105 = 3*5*7 with maximum order of primeness is 5, with order 3.

MATHEMATICA

Table[If[n==1, 0, Max@@(Length[NestWhileList[PrimePi, PrimePi[#], PrimeQ]]&/@FactorInteger[n][[All, 1]])], {n, 100}]

CROSSREFS

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A303431, A316470.

KEYWORD

allocated

nonn

AUTHOR

Gus Wiseman, Nov 24 2018

STATUS

approved

editing