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

1,3

Comments

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

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

Links

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

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A007814, A000010, A002110.

Partial sums of A023506.

Sequence in context: A347805 A088130 A046840 * A020486 A091428 A047499

Adjacent sequences: A057770 A057771 A057772 * A057774 A057775 A057776

Keyword

nonn

Author

Labos Elemer, Nov 02 2000

Status

approved