A077218 Sum of numbers of prime factors (counted with multiplicities) of numbers between n-th and (n+1)-st prime.

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, 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, 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

Offset: 1

Author

Amarnath Murthy, Nov 03 2002

Keywords

• nonn
• changed

Comments

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

Examples

			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. [corrected by _Harvey P. Dale_, May 25 2011]
		

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001222, A052297.

Programs

Mathematica

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

Formula

a(n) = Sum_{k=A000040(n)+1..A000040(n+1)-1} A001222(k). - Reinhard Zumkeller, Nov 29 2002

Extensions

More terms and better description from Reinhard Zumkeller, Nov 29 2002