A046645 a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).

0, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 7, 1, 4, 1, 4, 2, 2, 1, 5, 3, 2, 4, 4, 1, 3, 1, 8, 2, 2, 2, 6, 1, 2, 2, 5, 1, 3, 1, 4, 4, 2, 1, 8, 3, 4, 2, 4, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 4, 10, 2, 3, 1, 4, 2, 3, 1, 7, 1, 2, 4, 4, 2, 3, 1, 8, 7, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 4, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A268375 gives numbers n for which a(n) = A289617(n) = A005187(A001222(n)). - Antti Karttunen, Jul 08 2017

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65536
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

See A046643, A046644 for more details.

Cf. A001222, A005187, A007814, A077761, A268375, A289617, A289618, A293442, A293447.

Programs

Mathematica

f[p_, e_] := 2*e - DigitCount[2*e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

PARI

A007814(n) = (valuation(n,2));
A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(dA046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); } \\ After the Maple-program given in A046643.
A046645(n) = A007814(denominator(A046643perA046644(n))); \\ Antti Karttunen, Jul 08 2017

PARI

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046645(n) = vecsum(apply(e -> A005187(e), factorint(n)[, 2])); \\ A faster implementation. - Antti Karttunen, Jul 08 2017

Formula

a(n) = A007814(A046644(n)). - Michel Marcus, Apr 16 2015

Additive with a(p^n) = A005187(n). - Antti Karttunen, Jul 08 2017

a(n) = A293447(A293442(n)). - Antti Karttunen, Nov 10 2017

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 1.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 29 2023