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A077218
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Sum of numbers of prime factors (counted with multiplicities) of numbers between n-th and (n+1)-th prime.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
The number of elements in the (SFP) Smarandache Factor Partition of P(n) (product of composite numbers between successive primes) with maximum length.
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REFERENCES
| Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 2000.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
M. L. Perez et al., eds., Smarandache Notions Journal
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FORMULA
| sum{A001222(k): A000040(n)<k < A000040(n+1)}. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 29 2002
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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.
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MATHEMATICA
| Total[PrimeOmega[Range[First[#]+1, Last[#]-1]]]&/@Partition[Prime[ Range[90]], 2, 1] (* From Harvey P. Dale, May 25 2011 *)
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CROSSREFS
| Cf. A052297
Sequence in context: A029632 A089588 A014840 * A102780 A115025 A075428
Adjacent sequences: A077215 A077216 A077217 * A077219 A077220 A077221
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KEYWORD
| nonn
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 03 2002
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EXTENSIONS
| More terms and better description from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 29 2002
Corrected example [from Harvey P. Dale, May 25 2011]
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