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

1,3

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

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).

Prog

(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); };

Crossrefs

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

Sequence in context: A373516 A324754 A174220 * A048675 A162474 A334878

Adjacent sequences: A334868 A334869 A334870 * A334872 A334873 A334874

Keyword

nonn

Author

Antti Karttunen, Jun 08 2020

Status

approved