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A252464
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a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.
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66
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0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 5, 6, 9, 5, 4, 7, 4, 6, 10, 5, 11, 5, 6, 8, 5, 5, 12, 9, 7, 6, 13, 6, 14, 7, 5, 10, 15, 6, 5, 5, 8, 8, 16, 5, 6, 7, 9, 11, 17, 6, 18, 12, 6, 6, 7, 7, 19, 9, 10, 6, 20, 6, 21, 13, 5, 10, 6, 8, 22, 7, 5, 14, 23, 7, 8, 15, 11, 8, 24, 6, 7, 11, 12, 16, 9, 7, 25, 6, 7, 6, 26, 9, 27
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OFFSET
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1,3
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COMMENTS
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a(n) tells how many iterations of A252463 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A005940 and A163511.
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LINKS
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FORMULA
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a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).
Other identities. For all n >= 1:
And in general, a(prime(n)^k) = n+k-1.
a(A000079(n)) = n. [I.e., a(2^n) = n.]
For all n >= 2:
a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).
(End).
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EXAMPLE
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The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the inner lining of the integer partition with Heinz number n, which is also the size of the largest hook of the same partition. For example, the partition with Heinz number 715 is (6,5,3), with diagram
o o o o o o
o o o o o
o o o
which has inner lining
o o
o o o
o o o
and largest hook
o o o o o o
o
o
both of which have size 8, so a(715) = 8.
(End)
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MATHEMATICA
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Table[If[n==1, 1, PrimeOmega[n]+PrimePi[FactorInteger[n][[-1, 1]]]]-1, {n, 100}] (* Gus Wiseman, Apr 02 2019 *)
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PROG
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(Scheme, two different versions)
;; Memoization-macro definec can be found from Antti Karttunen's IntSeq-library
(PARI)
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
(Python)
from sympy import primepi, primeomega, primefactors
def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # Chai Wah Wu, Jul 17 2023
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CROSSREFS
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Cf. A000040, A000079, A001221, A001222, A005940, A029837, A061395, A156552 (A005941), A163511, A243071, A252461, A252463, A252735, A252736, A252759, A253553, A253563, A253565, A297113, A297155, A297167, A324870, A324872.
Right edge of irregular triangle A265146.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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