A057773 a(n) = Sum_{i=1..n} nu_2(prime(i) - 1) where nu_2(m) = exponent of highest power of 2 dividing m.

0, 1, 3, 4, 5, 7, 11, 12, 13, 15, 16, 18, 21, 22, 23, 25, 26, 28, 29, 30, 33, 34, 35, 38, 43, 45, 46, 47, 49, 53, 54, 55, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 78, 80, 81, 82, 83, 84, 86, 89, 90, 94, 95, 103, 104, 106, 107, 109, 112, 113, 115, 116, 117, 120, 122, 123

Offset: 1

Author

Labos Elemer, Nov 02 2000

Keywords

• nonn

Comments

Exponent of highest power of 2 dividing Euler phi of primorials.

Conjecture: a(n) ~ 2n. - Charles R Greathouse IV, Jun 02 2015

The conjecture is true. See the comment in A023506 about the asymptotic distribution of nu_2(prime(i) - 1). - Amiram Eldar, Mar 14 2025

Examples

			For n = 6, the 6th primorial is 30030, phi(30030) = 5760 = 2^7 * 3^2 * 5, so a(6) = 7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007814, A000010, A002110.

Partial sums of A023506.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)+padic[ordp](ithprime(n)-1, 2))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 01 2023

Mathematica

Table[IntegerExponent[EulerPhi[Product[Prime[i], {i, n}]], 2], {n, 110}] (* Jamie Morken, Oct 13 2023 *)

PARI

a(n) = sum(k=1,n, valuation(prime(k)-1, 2)); \\ Michel Marcus, May 30 2015

PARI

a(n) = valuation(eulerphi(prod(k=1,n, prime(k))), 2); \\ Michel Marcus, May 30 2015

PARI

first(n)=my(p=primes(n),s); vector(#p,i,s+=valuation(p[i]-1,2)) \\ Charles R Greathouse IV, Jun 02 2015

Formula

a(n) = A007814(A000010(A002110(n))).