A051282 2-adic valuation of A025487: largest k such that 2^k divides A025487(n), where A025487 gives products of primorials.

0, 1, 2, 1, 3, 2, 4, 3, 1, 5, 2, 4, 2, 6, 3, 5, 3, 7, 4, 2, 6, 1, 3, 4, 8, 5, 3, 7, 2, 4, 5, 9, 6, 4, 8, 3, 5, 2, 6, 10, 3, 7, 2, 4, 5, 9, 4, 6, 3, 7, 11, 4, 8, 1, 3, 5, 6, 10, 5, 7, 4, 8, 12, 5, 9, 2, 4, 6, 3, 7, 11, 2, 4, 6, 8, 5, 3, 9, 5, 13, 6, 10, 3, 5, 7, 4, 8, 12, 3, 5, 7, 9, 2, 6, 4, 10, 6, 14, 7, 11, 4, 6, 8, 5, 9, 13, 4, 6, 8, 3, 10, 3, 7, 1, 5, 11, 7, 4

Offset

1,3

Comments

a(n) can be used for resorting A025487 and sequences indexed by A025487, e.g., A050322, A050323, A050324 and A050325.

a(n) is the number of primorial numbers (A002110) larger than 1 in the representation of A025487(n) as a product of primorial numbers. - Amiram Eldar, Jun 03 2023

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = A007814(A025487(n)) = A051903(A025487(n)). - Matthew Vandermast, Jul 03 2012

Example

a(8) = 3 because A025487(8) = 24 and 2^3 divides 24.

Mathematica

max = 40000; A025487 = {1}; lpe = {}; Do[ pe = Sort[ FactorInteger[n][[All, 2]]]; If[FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[A025487, n]], {n, 2, max}]; a[n_] := FactorInteger[ A025487[[n]] ][[1, 2]]; a[1] = 0; Table[a[n], {n, 1, Length[A025487]}] (* Jean-François Alcover, Jun 14 2012, after Robert G. Wilson v *)

Prog

(Haskell)
a051282 = a007814 . a025487  -- Reinhard Zumkeller, Apr 06 2013
(PARI) isA025487(n)=my(k=valuation(n, 2), t); n>>=k; forprime(p=3, default(primelimit), t=valuation(n, p); if(t>k, return(0), k=t); if(k, n/=p^k, return(n==1)))
[valuation(n, 2) | n <- [1..1000], isA025487(n)]
\\ Or, for older versions:
apply(n->valuation(n, 2), select(isA025487, [1..1000])) \\ Charles R Greathouse IV, Nov 07 2014
(Python)
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log
from oeis_sequences.OEISsequences import bisection
def A051282(n):
    @lru_cache(maxsize=None)
    def g(x, m, j): return sum(g(x//(prime(m)**i), m-1, i) for i in range(j, integer_log(x, prime(m))[0]+1)) if m-1 else max(0, x.bit_length()-j)
    def f(x):
        c, p = n-1+x, 1
        for k in count(1):
            p *= prime(k)
            if p>x:
                break
            c -= g(x, k, 1)
        return c
    return (~(m:=bisection(f, n, n)) & m-1).bit_length() # Chai Wah Wu, Apr 08 2026

Crossrefs

Cf. A001055, A002033, A007814, A025487, A045778, A050320, A051903.

Cf. A050322, A050323, A050324, A050325.

Sequence in context: A285120 A282744 A195310 * A274121 A052306 A322886

Adjacent sequences: A051279 A051280 A051281 * A051283 A051284 A051285

Keyword

nice,nonn

Author

Alford Arnold

Extensions

More terms from Naohiro Nomoto, Mar 11 2001

Status

approved