%I #21 Apr 22 2019 22:15:39

%S 0,1,2,2,3,2,4,3,3,3,5,3,6,4,3,4,7,3,8,3,4,5,9,4,4,6,4,4,10,3,11,5,5,

%T 7,4,4,12,8,6,4,13,4,14,5,4,9,15,5,5,4,7,6,16,4,5,4,8,10,17,4,18,11,4,

%U 6,6,5,19,7,9,4,20,5,21,12,4,8,5,6,22,5,5,13,23,4,7,14,10,5,24,4,6,9,11

%N Number of prime cascades to reach 1, where a prime cascade (A065769) is multiplicative with a(p(m)^k) = p(m-1) * p(m)^(k-1).

%C 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

%C 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

%H Antti Karttunen, <a href="/A065770/b065770.txt">Table of n, a(n) for n = 1..16384</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000384">St000384: The maximal part of the shifted composition of an integer partition</a>

%F Inverse of primes, powers of 2 and primorials in sense that a(A000040(n))=n; a(A000079(n))=n; a(A002110(n))=n. If n>0: a(3^n)=n+1; a(2^n*3^k)=n+k; a(p(k)^n)=n+k-1; a(n!)=A022559(n).

%F a(1) = 0; and for n > 1, a(n) = 1 + A065769(n). - _Antti Karttunen_, Dec 31 2017

%e 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.

%e From _Gus Wiseman_, Apr 06 2019: (Start)

%e The partition with Heinz number 7865 is (6,5,5,3), with diagram

%e o o o o o o

%e o o o o o

%e o o o o o

%e o o o

%e which has longest path from (1,1) to (5,3) of length 6, so a(7865) = 7.

%e (End)

%t 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 *)

%o (Scheme) (definec (A065770 n) (if (= 1 n) 0 (+ 1 (A065770 (A065769 n))))) ;; _Antti Karttunen_, Dec 31 2017

%Y Cf. A065769.

%Y Differs from A297113 for the first time at n=20, where a(20) = 3, while A297113(20) = 4.

%Y Cf. A001221, A001222, A052126, A056239, A064989, A112798, A174090, A257990, A325166, A325167, A325169.

%K nonn

%O 1,3

%A _Henry Bottomley_, Nov 19 2001