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

1,2

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.

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

Formula

sum{A001222(k): A000040(n)<k < A000040(n+1)}. - Reinhard Zumkeller, Nov 29 2002

Example

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.

Mathematica

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

Crossrefs

Cf. A052297

Sequence in context: A341760 A342453 A370393 * A102780 A227828 A115025

Adjacent sequences: A077215 A077216 A077217 * A077219 A077220 A077221

Keyword

nonn

Author

Amarnath Murthy, Nov 03 2002

Extensions

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

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

Status

approved