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Exponent of highest power of 2 dividing the product of the composite numbers between the n-th prime and the (n+1)-st prime.
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%I #62 Aug 12 2024 19:26:27

%S 0,2,1,4,2,5,1,3,6,1,8,4,1,3,7,5,2,8,3,3,4,5,6,9,3,1,4,2,5,11,8,6,1,

%T 10,1,6,7,3,6,6,2,8,6,3,1,12,10,6,2,4,4,4,8,11,4,6,1,7,4,1,11,13,3,3,

%U 3,15,7,8,2,6,4,7,7,5,3,10,7,5,7

%N Exponent of highest power of 2 dividing the product of the composite numbers between the n-th prime and the (n+1)-st prime.

%H Robert Israel, <a href="/A276133/b276133.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A007814(A061214(n)).

%F a(n+1) = Sum_{k = A000040(n+1)..A000040(n+2)} A007814(k).

%p A:= Vector(100): q:= 2:

%p for n from 1 to 100 do

%p p:= q; q:= nextprime(q);

%p t:= 0;

%p for i from p+1 to q-1 do t:= t + padic:-ordp(i,2) od;

%p A[n]:= t

%p od:

%p convert(A,list); # _Robert Israel_, Apr 11 2021

%t IntegerExponent[#,2]&/@(Times@@Range[#[[1]]+1,#[[2]]-1]&/@Partition[ Prime[ Range[ 80]],2,1]) (* _Harvey P. Dale_, Aug 12 2024 *)

%o (PARI) a(n) = valuation(prod(k=prime(n)+1, prime(n+1)-1, k), 2); \\ _Michel Marcus_, Aug 31 2016

%o (PARI) a(n) = my(p=prime(n+1),q=prime(n)); p-hammingweight(p) - (q-hammingweight(q)); \\ _Kevin Ryde_, Apr 11 2021

%o (Python)

%o from sympy import prime

%o def A276133(n): return (p:=prime(n+1)-1)-p.bit_count()-(q:=prime(n))+q.bit_count() # _Chai Wah Wu_, Jul 10 2022

%Y Supersequence of A205649 (Hamming distance between twin primes).

%Y Cf. A000040, A007814, A061214.

%Y First differences of A080085.

%K nonn,easy

%O 1,2

%A _Juri-Stepan Gerasimov_, Aug 29 2016

%E a(16) corrected by _Robert Israel_, Apr 11 2021