A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)).

0, 1, 2, 2, 3, 2, 4, 3, 3, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 4, 4, 5, 9, 4, 4, 6, 4, 5, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 5, 13, 4, 14, 6, 4, 9, 15, 5, 5, 4, 7, 7, 16, 4, 5, 6, 8, 10, 17, 4, 18, 11, 5, 6, 6, 5, 19, 8, 9, 4, 20, 5, 21, 12, 4, 9, 5, 6, 22, 6, 5, 13, 23, 5, 7, 14, 10, 7, 24, 4, 6, 10, 11, 15, 8, 6, 25

Offset

1,3

Comments

From Gus Wiseman, Apr 06 2019: (Start)

Also the number of squares in the Young diagram of the integer partition with Heinz number n that are graph-distance 1 from the lower-right boundary, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (6,5,5,3) with Heinz number 7865 has diagram

o o o o o o

o o o o o

o o o o o

o o o

with inner rim

o

o

o o

o o o

of size 7, so a(7867) = 7.

(End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..12721

Index entries for sequences computed from indices in prime factorization

Formula

a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)) .

For n > 1, a(n) = A001511(A297112(n)), where A297112(n) = Sum_{d|n} moebius(n/d)*A156552(d).

a(n) = A252464(n) - A297155(n).

For n > 1, a(n) = 1+A033265(A156552(n)) = 1+A297167(n) = A046660(n) + A061395(n). - Last two sums added by Antti Karttunen, Sep 02 2018

Other identities. For all n >= 1:

a(A000040(n)) = n. [Each n occurs for the first time at the n-th prime.]

Mathematica

Table[If[n==1, 0, PrimePi[FactorInteger[n][[-1, 1]]]+PrimeOmega[n]-PrimeNu[n]], {n, 100}] (* Gus Wiseman, Apr 06 2019 *)

Prog

(PARI)
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
A297113(n) = if(n<=2, n-1, if(n%2, 1+A297113(A064989(n)), !(n%4)+A297113(n/2)));
\\ More complex way, after Moebius transform:
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n, d, moebius(n/d)*A156552(d));
A297113(n) = if(1==n, 0, 1+valuation(A297112(n), 2));
(Scheme, with memoization-macro definec)
(definec (A297113 n) (cond ((<= n 2) (- n 1)) ((= 2 (modulo n 4)) (A297113 (/ n 2))) (else (+ 1 (A297113 (A252463 n))))))

Crossrefs

One more than A297167 (after the initial term).

Cf. A001511, A008683, A033265, A064989, A156552, A252463, A252464, A297112, A297155.

Cf. also A297157, A297161, A297162.

Cf. A052126, A065770, A112798, A115994, A174090, A325166, A325167, A325169.

Sequence in context: A155043 A337327 A065770 * A086375 A107324 A023522

Adjacent sequences: A297110 A297111 A297112 * A297114 A297115 A297116

Keyword

nonn

Author

Antti Karttunen, Dec 26 2017

Status

approved