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.

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

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 29 2016

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Supersequence of A205649 (Hamming distance between twin primes).

Cf. A000040, A007814, A061214.

First differences of A080085.

Programs

Maple

A:= Vector(100): q:= 2:
for n from 1 to 100 do
  p:= q; q:= nextprime(q);
  t:= 0;
  for i from p+1 to q-1 do t:= t + padic:-ordp(i,2) od;
  A[n]:= t
od:
convert(A,list); # Robert Israel, Apr 11 2021

Mathematica

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

PARI

a(n) = valuation(prod(k=prime(n)+1, prime(n+1)-1, k), 2); \\ Michel Marcus, Aug 31 2016

PARI

a(n) = my(p=prime(n+1),q=prime(n)); p-hammingweight(p) - (q-hammingweight(q)); \\ Kevin Ryde, Apr 11 2021

Python

from sympy import prime
def A276133(n): return (p:=prime(n+1)-1)-p.bit_count()-(q:=prime(n))+q.bit_count() # Chai Wah Wu, Jul 10 2022

Formula

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

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

Extensions

a(16) corrected by Robert Israel, Apr 11 2021