cp's OEIS Frontend

A252464 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.

%I A252464 #45 Jul 17 2023 17:07:01
%S A252464 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,
%T A252464 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,
%U A252464 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
%N A252464 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.
%C A252464 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.
%C A252464 Similarly for A253553 in trees A253563 and A253565. - _Antti Karttunen_, Apr 14 2019
%H A252464 Antti Karttunen, <a href="/A252464/b252464.txt">Table of n, a(n) for n = 1..8192</a>
%F A252464 a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).
%F A252464 a(n) = A029837(1+A243071(n)). [a(n) = binary width of terms of A243071.]
%F A252464 a(n) = A029837(A005941(n)) = A029837(1+A156552(n)). [Also binary width of terms of A156552.]
%F A252464 Other identities. For all n >= 1:
%F A252464 a(A000040(n)) = n.
%F A252464 a(A001248(n)) = n+1.
%F A252464 a(A030078(n)) = n+2.
%F A252464 And in general, a(prime(n)^k) = n+k-1.
%F A252464 a(A000079(n)) = n. [I.e., a(2^n) = n.]
%F A252464 For all n >= 2:
%F A252464 a(n) = A001222(n) + A061395(n) - 1 = A001222(n) + A252735(n) = A061395(n) + A252736(n) = 1 + A252735(n) + A252736(n).
%F A252464 a(n) = A325134(n) - 1. - _Gus Wiseman_, Apr 02 2019
%F A252464 From _Antti Karttunen_, Apr 14 2019: (Start)
%F A252464 a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).
%F A252464 a(n) = A001221(n) + A297167(n) = A297113(n) + A297155(n).
%F A252464 (End).
%e A252464 From _Gus Wiseman_, Apr 02 2019: (Start)
%e A252464 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
%e A252464   o o o o o o
%e A252464   o o o o o
%e A252464   o o o
%e A252464 which has inner lining
%e A252464           o o
%e A252464       o o o
%e A252464   o o o
%e A252464 and largest hook
%e A252464   o o o o o o
%e A252464   o
%e A252464   o
%e A252464 both of which have size 8, so a(715) = 8.
%e A252464 (End)
%t A252464 Table[If[n==1,1,PrimeOmega[n]+PrimePi[FactorInteger[n][[-1,1]]]]-1,{n,100}] (* _Gus Wiseman_, Apr 02 2019 *)
%o A252464 (Scheme, two different versions)
%o A252464 ;; Memoization-macro definec can be found from _Antti Karttunen_'s IntSeq-library
%o A252464 (definec (A252464 n) (if (<= n 1) 0 (+ 1 (A252464 (A252463 n)))))
%o A252464 (define (A252464 n) (A029837 (+ 1 (A243071 n))))
%o A252464 (define (A252464 n) (A029837 (A005941 n)))
%o A252464 (PARI)
%o A252464 A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
%o A252464 A252464(n) = (bigomega(n) + A061395(n) - 1); \\ _Antti Karttunen_, Apr 14 2019
%o A252464 (Python)
%o A252464 from sympy import primepi, primeomega, primefactors
%o A252464 def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # _Chai Wah Wu_, Jul 17 2023
%Y A252464 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.
%Y A252464 Right edge of irregular triangle A265146.
%Y A252464 Cf. also A246348.
%Y A252464 Cf. A052126, A056239, A093641, A112798, A257990, A325133, A325134, A325135.
%K A252464 nonn
%O A252464 1,3
%A A252464 _Antti Karttunen_, Dec 20 2014