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A276133 Exponent of highest power of 2 dividing the product of the composite numbers between the n-th prime and the (n+1)-st prime.

%I A276133 #62 Aug 12 2024 19:26:27
%S A276133 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 A276133 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 A276133 3,15,7,8,2,6,4,7,7,5,3,10,7,5,7
%N A276133 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 A276133 Robert Israel, <a href="/A276133/b276133.txt">Table of n, a(n) for n = 1..10000</a>
%F A276133 a(n) = A007814(A061214(n)).
%F A276133 a(n+1) = Sum_{k = A000040(n+1)..A000040(n+2)} A007814(k).
%p A276133 A:= Vector(100): q:= 2:
%p A276133 for n from 1 to 100 do
%p A276133   p:= q; q:= nextprime(q);
%p A276133   t:= 0;
%p A276133   for i from p+1 to q-1 do t:= t + padic:-ordp(i,2) od;
%p A276133   A[n]:= t
%p A276133 od:
%p A276133 convert(A,list); # _Robert Israel_, Apr 11 2021
%t A276133 IntegerExponent[#,2]&/@(Times@@Range[#[[1]]+1,#[[2]]-1]&/@Partition[ Prime[ Range[ 80]],2,1]) (* _Harvey P. Dale_, Aug 12 2024 *)
%o A276133 (PARI) a(n) = valuation(prod(k=prime(n)+1, prime(n+1)-1, k), 2); \\ _Michel Marcus_, Aug 31 2016
%o A276133 (PARI) a(n) = my(p=prime(n+1),q=prime(n)); p-hammingweight(p) - (q-hammingweight(q)); \\ _Kevin Ryde_, Apr 11 2021
%o A276133 (Python)
%o A276133 from sympy import prime
%o A276133 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 A276133 Supersequence of A205649 (Hamming distance between twin primes).
%Y A276133 Cf. A000040, A007814, A061214.
%Y A276133 First differences of A080085.
%K A276133 nonn,easy
%O A276133 1,2
%A A276133 _Juri-Stepan Gerasimov_, Aug 29 2016
%E A276133 a(16) corrected by _Robert Israel_, Apr 11 2021