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a(n) = Sum_{i=1..n} nu_2(prime(i) - 1) where nu_2(m) = exponent of highest power of 2 dividing m.
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%I #34 Oct 13 2023 16:34:59

%S 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,

%T 45,46,47,49,53,54,55,58,59,61,62,64,65,66,68,69,71,72,78,80,81,82,83,

%U 84,86,89,90,94,95,103,104,106,107,109,112,113,115,116,117,120,122,123

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

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

%C Conjecture: a(n) ~ 2n. - _Charles R Greathouse IV_, Jun 02 2015

%H Charles R Greathouse IV, <a href="/A057773/b057773.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%p a:= proc(n) option remember; `if`(n<2, 0,

%p a(n-1)+padic[ordp](ithprime(n)-1, 2))

%p end:

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Jan 01 2023

%t Table[IntegerExponent[EulerPhi[Product[Prime[i], {i, n}]], 2], {n, 110}] (* _Jamie Morken_, Oct 13 2023 *)

%o (PARI) a(n) = sum(k=1,n, valuation(prime(k)-1, 2)); \\ _Michel Marcus_, May 30 2015

%o (PARI) a(n) = valuation(eulerphi(prod(k=1,n, prime(k))), 2); \\ _Michel Marcus_, May 30 2015

%o (PARI) first(n)=my(p=primes(n),s); vector(#p,i,s+=valuation(p[i]-1,2)) \\ _Charles R Greathouse IV_, Jun 02 2015

%Y Cf. A007814, A000010, A002110.

%Y Partial sums of A023506.

%K nonn

%O 1,3

%A _Labos Elemer_, Nov 02 2000