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

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

Offset: 1

Author

Antti Karttunen, Jun 08 2020

Keywords

• nonn

Comments

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

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

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).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

PARI

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));
A334871(n) = { my(s=0); while(n>1,s++; n = A334870(n)); (s); };

PARI

\\ Much faster:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A334871(n) = { my(s=0); while(n>1, if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

Formula

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

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

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

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

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

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

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