OFFSET
1,3
COMMENTS
It seems that a(n) <= A297113(n) for all n. Of the first 10000 positive natural numbers, 6454 are such that a(n) = A297113(n). - Antti Karttunen, Dec 31 2017
Also one plus the maximum number of unit steps East or South in the Young diagram of the integer partition with Heinz number n > 1, starting from the upper-left square and ending in a boundary square in the lower-right quadrant. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 06 2019
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
EXAMPLE
a(50) = 4 since the cascade goes from 50 = 2^1 * 5^2 to 15 = 3^1 * 5^1 to 6 = 2^1 * 3^1 to 2 = 2^1 to 1.
From Gus Wiseman, Apr 06 2019: (Start)
The partition with Heinz number 7865 is (6,5,5,3), with diagram
o o o o o o
o o o o o
o o o o o
o o o
which has longest path from (1,1) to (5,3) of length 6, so a(7865) = 7.
(End)
MATHEMATICA
Table[If[n==1, 0, Max@@Total/@Position[PadRight[ConstantArray[1, #]&/@Sort[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]], Greater]], 1]-1], {n, 100}] (* Gus Wiseman, Apr 06 2019 *)
PROG
(Scheme) (definec (A065770 n) (if (= 1 n) 0 (+ 1 (A065770 (A065769 n))))) ;; Antti Karttunen, Dec 31 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, Nov 19 2001
STATUS
approved