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

%I A057773 #39 Mar 14 2025 04:34:45
%S A057773 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 A057773 45,46,47,49,53,54,55,58,59,61,62,64,65,66,68,69,71,72,78,80,81,82,83,
%U A057773 84,86,89,90,94,95,103,104,106,107,109,112,113,115,116,117,120,122,123
%N A057773 a(n) = Sum_{i=1..n} nu_2(prime(i) - 1) where nu_2(m) = exponent of highest power of 2 dividing m.
%C A057773 Exponent of highest power of 2 dividing Euler phi of primorials.
%C A057773 Conjecture: a(n) ~ 2n. - _Charles R Greathouse IV_, Jun 02 2015
%C A057773 The conjecture is true. See the comment in A023506 about the asymptotic distribution of nu_2(prime(i) - 1). - _Amiram Eldar_, Mar 14 2025
%H A057773 Charles R Greathouse IV, <a href="/A057773/b057773.txt">Table of n, a(n) for n = 1..10000</a>
%F A057773 a(n) = A007814(A000010(A002110(n))).
%e A057773 For n = 6, the 6th primorial is 30030, phi(30030) = 5760 = 2^7 * 3^2 * 5, so a(6) = 7.
%p A057773 a:= proc(n) option remember; `if`(n<2, 0,
%p A057773       a(n-1)+padic[ordp](ithprime(n)-1, 2))
%p A057773     end:
%p A057773 seq(a(n), n=1..80);  # _Alois P. Heinz_, Jan 01 2023
%t A057773 Table[IntegerExponent[EulerPhi[Product[Prime[i], {i, n}]], 2], {n, 110}] (* _Jamie Morken_, Oct 13 2023 *)
%o A057773 (PARI) a(n) = sum(k=1,n, valuation(prime(k)-1, 2)); \\ _Michel Marcus_, May 30 2015
%o A057773 (PARI) a(n) = valuation(eulerphi(prod(k=1,n, prime(k))), 2); \\ _Michel Marcus_, May 30 2015
%o A057773 (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 A057773 Cf. A007814, A000010, A002110.
%Y A057773 Partial sums of A023506.
%K A057773 nonn
%O A057773 1,3
%A A057773 _Labos Elemer_, Nov 02 2000