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A123369
Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).
1
0, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 2, 2, 3, 2, 4, 3, 1, 2, 5, 3, 3, 2, 2, 1, 3, 3, 3, 1, 1, 2, 2, 2, 5, 2, 2, 2, 5, 1, 3, 4, 4, 3, 3, 3, 5, 4, 3, 3, 3, 2, 2, 6, 2, 3, 4, 2, 4, 2, 3, 3, 2, 4, 4, 4, 3, 3, 3, 3, 4, 2, 3, 4, 2, 2, 5, 3, 2, 2, 4, 4, 2, 2, 1, 6, 4, 2, 5, 3, 5, 1, 2, 2, 3, 4, 2, 3, 3, 3, 5
OFFSET
1,4
COMMENTS
We must include multiplicity in the definition due to terms such as a(16) = 29889983 = 19 * 31^2 * 1637. The primes are those n for which a(n) = Omega(A027612(n))= 1, namely a(2) = 5, a(3) = 13, a(6) = 223, a(9) = 4861, a(18) = 197698279, a(25) = 25472027467. The semiprimes are those for which a(n) = 2, such as when n = 4, 5, 7, 8, 11, 12, 13, 15, 19, 23, 24. The 3-almost primes are those for which a(n) = 3, as with the "3-brilliant" a(10) = 55991 = 13 * 59 * 73, a(14), a(17), a(21), a(22), a(26).
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, pp. 143 and 258-259.
LINKS
Eric Weisstein's World of Mathematics, Harmonic Number, MathWorld, see discussion of Conway and Guy (1996) definition of the second-order harmonic number.
FORMULA
a(n) = A001222(A027612(n)) = Omega(Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n).
EXAMPLE
a(20) = 5 because A027612(20) = 41054655 = 3 * 5 * 23 * 127 * 937 has 5 prime factors.
MATHEMATICA
PrimeOmega[Numerator[Table[Sum[k/(n - k + 1), {k, 1, n}], {n, 1, 50}]]] (* G. C. Greubel, Jan 22 2017 *)
CROSSREFS
Cf. A001222 Number of prime divisors of n (counted with multiplicity), A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n, A027611, A001008, A002805, A001705, A006675, A093418.
Sequence in context: A069163 A025260 A227156 * A178306 A309363 A280153
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Nov 09 2006
STATUS
approved