A334871 - OEIS

%I #18 Jun 09 2020 22:12:14

%S 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 512,7,1024,4,18,65,12,4,2048,129,34,7,4096,11,8192,18,7,257,16384,5,

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

%C Distance of n from the root (1) in binary trees like A334860 and A334866.

%C Each n > 0 occurs 2^(n-1) times.

%C 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 Antti Karttunen, <a href="/A334871/b334871.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 0; for n > 1, a(n) = 1 + a(A334870(n)).

%F a(n) = A252464(A225546(n)).

%F a(n) = A048675(A007913(n)) + a(A008833(n)).

%F For n > 1, a(n) = 1 + A048675(A007913(n)) + a(A000188(n)).

%F For n > 1, a(n) = A070939(A334859(n)) = A070939(A334865(n)).

%F For all n >= 1, a(n) >= A299090(n).

%F For all n >= 1, a(n) >= A334872(n).

%o (PARI)

%o 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(n) = { my(s=0); while(n>1,s++; n = A334870(n)); (s); };

%o (PARI)

%o \\ Much faster:

%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

%o A334871(n) = { my(s=0); while(n>1, if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

%Y Cf. A007913, A008833, A070939, A225546, A252464, A299090, A334859, A334860, A334865, A334866, A334869, A334870, A334872.

%K nonn

%O 1,3

%A _Antti Karttunen_, Jun 08 2020