cp's OEIS Frontend

A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

%I A334871 #18 Jun 09 2020 22:12:14
%S A334871 0,1,2,2,4,3,8,3,3,5,16,4,32,9,6,3,64,4,128,6,10,17,256,5,5,33,5,10,
%T A334871 512,7,1024,4,18,65,12,4,2048,129,34,7,4096,11,8192,18,7,257,16384,5,
%U A334871 9,6,66,34,32768,6,20,11,130,513,65536,8,131072,1025,11,4,36,19,262144,66,258,13,524288,5,1048576,2049,7,130
%N A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.
%C A334871 Distance of n from the root (1) in binary trees like A334860 and A334866.
%C A334871 Each n > 0 occurs 2^(n-1) times.
%C A334871 a(n) is the size of the inner lining of the integer partition with Heinz number A225546(n), which is also the size of the largest hook of the same partition. (After _Gus Wiseman_'s Apr 02 2019 comment in A252464).
%H A334871 Antti Karttunen, <a href="/A334871/b334871.txt">Table of n, a(n) for n = 1..10000</a>
%F A334871 a(1) = 0; for n > 1, a(n) = 1 + a(A334870(n)).
%F A334871 a(n) = A252464(A225546(n)).
%F A334871 a(n) = A048675(A007913(n)) + a(A008833(n)).
%F A334871 For n > 1, a(n) = 1 + A048675(A007913(n)) + a(A000188(n)).
%F A334871 For n > 1, a(n) = A070939(A334859(n)) = A070939(A334865(n)).
%F A334871 For all n >= 1, a(n) >= A299090(n).
%F A334871 For all n >= 1, a(n) >= A334872(n).
%o A334871 (PARI)
%o A334871 A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
%o A334871 A334871(n) = { my(s=0); while(n>1,s++; n = A334870(n)); (s); };
%o A334871 (PARI)
%o A334871 \\ Much faster:
%o A334871 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A334871 A334871(n) = { my(s=0); while(n>1, if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };
%Y A334871 Cf. A007913, A008833, A070939, A225546, A252464, A299090, A334859, A334860, A334865, A334866, A334869, A334870, A334872.
%K A334871 nonn
%O A334871 1,3
%A A334871 _Antti Karttunen_, Jun 08 2020