%I #17 Jul 02 2017 09:54:38

%S 0,2,2,7,3,8,3,7,14,3,15,8,3,8,15,14,4,16,8,5,13,11,14,21,10,3,9,5,10,

%T 36,12,16,3,26,4,16,17,8,16,15,5,26,7,9,4,33,30,12,4,10,14,6,29,20,14,

%U 15,5,17,10,3,28,40,9,5,9,42,16,27,4,14,13,22,17,18,8,19,22,11,23,27,5

%N Sum of numbers of prime factors (counted with multiplicities) of numbers between n-th and (n+1)-st prime.

%C Also, number of prime factors (with multiplicity) of the product P(n) of the composite numbers between n-th and (n+1)-th prime.

%D Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 2000.

%H T. D. Noe, <a href="/A077218/b077218.txt">Table of n, a(n) for n = 1..10000</a>

%F sum{A001222(k): A000040(n)<k < A000040(n+1)}. - _Reinhard Zumkeller_, Nov 29 2002

%e a(6) = 8. Prime(6) = 13 and prime(7) = 17. 14, 15, and 16 are the composite numbers between 13 and 17. 14 has two prime factors (2 and 7); 15 has two prime factors (3 and 5); and 16 has four prime factors (2, 2, 2, and 2). Thus, a(6) = 2 + 2 + 4 = 8 total prime factors.

%t Total[PrimeOmega[Range[First[#]+1,Last[#]-1]]]&/@Partition[Prime[ Range[90]],2,1] (* _Harvey P. Dale_, May 25 2011 *)

%Y Cf. A052297

%K nonn

%O 1,2

%A _Amarnath Murthy_, Nov 03 2002

%E More terms and better description from _Reinhard Zumkeller_, Nov 29 2002

%E Corrected example [from Harvey P. Dale, May 25 2011]