A140122
Negative of numerator of Sum_{k=1..n} (-1)^k / semiprime(k).
Original entry on oeis.org
1, 1, 7, 17, 209, 25, 37, 281, 9797, 92711, 120011, 1589737, 2027317, 30861373, 38322673, 735926129, 6107595203, 5188977503, 6040786643, 5218865543, 174771852097, 4738609625857, 5386574286277, 4776172794577, 197777244862999
Offset: 1
The first 10 values of a(n)/A140123(n) = -1/4, -1/12, -7/36, -17/180, -209/1260, -25/252, -37/252, -281/2772, -9797/69300, -92711/900900. The 10th term of the sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26) = -92711/900900 hence a(10) = -(-92711) = 92711. The 20th term of the alternating sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26)-(1/33)+(1/34)-(1/35)+(1/38)-(1/39)+(1/46)-(1/49)+(1/51)-(1/55)+(1/57) = -5218865543/46849502700, hence a(20) = 5218865543.
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A001358 := proc(n) local a; if n = 1 then 4; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A140122 := proc(n) local k ; numer(-add ( (-1)^k/A001358(k),k=1..n)) ; end: seq(A140122(n),n=1..30) ; # R. J. Mathar, May 13 2008
A191645
Numerators of the n-th partial "harmonic" sum of 1 + inverse semiprimes.
Original entry on oeis.org
5, 17, 55, 293, 2141, 445, 457, 5153, 131597, 1745411, 1772711, 30586537, 31024117, 597115873, 604577173, 14050770329, 99311504603, 100230122303, 101081931443, 101903852543
Offset: 1
a(1) = 5 because 1 + 1/4 = 5/4.
a(2) = 17 because 1 + 1/4 + 1/6 = 17/12.
a(3) = 55 because 1 + 1/4 + 1/6 + 1/9 = 55/36.
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A191645 := proc(n) 1+add(1/A001358(i),i=1..n) ; numer(%) ; end proc:
seq(A191645(n),n=1..20) ; # R. J. Mathar, Jun 16 2011
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With[{sp=Join[{1},Select[Range[100],PrimeOmega[#]==2&]]},Rest[ Numerator[ Accumulate[1/sp]]]] (* Harvey P. Dale, May 01 2015 *)
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s=1; for(k=1,99, bigomega(k)==2 & print1(numerator(s+=1/k)", ")) \\ M. F. Hasler, Jun 17 2011
Showing 1-2 of 2 results.
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