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A065770
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).
42
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, 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, 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
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
FORMULA
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).
a(1) = 0; and for n > 1, a(n) = 1 + A065769(n). - Antti Karttunen, Dec 31 2017
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
Cf. A065769.
Differs from A297113 for the first time at n=20, where a(20) = 3, while A297113(20) = 4.
Sequence in context: A327664 A155043 A337327 * A297113 A086375 A107324
KEYWORD
nonn
AUTHOR
Henry Bottomley, Nov 19 2001
STATUS
approved