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A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

%I A123369 #10 Feb 16 2025 08:33:03
%S A123369 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,
%T A123369 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,
%U A123369 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
%N A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).
%C A123369 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).
%D A123369 J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, pp. 143 and 258-259.
%H A123369 G. C. Greubel, <a href="/A123369/b123369.txt">Table of n, a(n) for n = 1..150</a>
%H A123369 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>, MathWorld, see discussion of Conway and Guy (1996) definition of the second-order harmonic number.
%F A123369 a(n) = A001222(A027612(n)) = Omega(Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n).
%e A123369 a(20) = 5 because A027612(20) = 41054655 = 3 * 5 * 23 * 127 * 937 has 5 prime factors.
%t A123369 PrimeOmega[Numerator[Table[Sum[k/(n - k + 1), {k, 1, n}], {n, 1, 50}]]] (* _G. C. Greubel_, Jan 22 2017 *)
%Y A123369 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.
%K A123369 easy,nonn
%O A123369 1,4
%A A123369 _Jonathan Vos Post_, Nov 09 2006