The On-Line Encyclopedia of Integer Sequences (OEIS)

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A278047

Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k*k'*(k+k')), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).
(history; published version)

#14 by Bruno Berselli at Wed Nov 23 02:49:24 EST 2016

STATUS

reviewed

approved

#13 by G. C. Greubel at Wed Nov 23 00:42:09 EST 2016

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proposed

reviewed

#12 by Michel Marcus at Wed Nov 23 00:15:31 EST 2016

STATUS

editing

proposed

#11 by Michel Marcus at Wed Nov 23 00:15:27 EST 2016

REFERENCES

J. Lehner and M. Newman, "Sums involving Farey fractions." Acta Arithmetica 15.2 (1969): 181-187.

LINKS

J. Lehner and M. Newman, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa15/aa15114.pdf">Sums involving Farey fractions</a>, Acta Arithmetica 15.2 (1969): 181-187.

STATUS

approved

editing

#10 by N. J. A. Sloane at Tue Nov 22 23:22:11 EST 2016

STATUS

editing

approved

#9 by N. J. A. Sloane at Tue Nov 22 23:22:09 EST 2016

REFERENCES

Newman, M., and J. Lehner and M. Newman, "Sums involving Farey fractions." Acta Arithmetica 15.2 (1969): 181-187.

STATUS

approved

editing

#8 by N. J. A. Sloane at Tue Nov 22 23:08:24 EST 2016

STATUS

editing

approved

#7 by N. J. A. Sloane at Tue Nov 22 23:08:22 EST 2016

CROSSREFS

Cf. A006843, A005728, A240877, A278046, A278048.

STATUS

approved

editing

#6 by N. J. A. Sloane at Tue Nov 22 23:07:43 EST 2016

STATUS

editing

approved

#5 by N. J. A. Sloane at Tue Nov 22 23:07:41 EST 2016

NAME

Let v = list of denominators of Farey series of order n (see A006843); alet b(n) = sum of products Sum 1/(k*k'*(k+k')), where (k,k') are pairs of adjacent successive terms of v; a(n) = numerator of b(n).

COMMENTS

Note the the sum of the reciprocals of these products is 1.

EXAMPLE

When n = 4, v = [1,4,3,2,3,4,1], so a(4) = 1*4 + 4*3 + 3*2 + 2*3 + 3*4 + 4*1 = 44.

The fractions b(n) are 1/2, 1/3, 7/30, 4/21, 37/252, 53/396, 707/6435, 85/858, 179077/2042040, 289613/3527160, 379721/5290740, 641671/9360540, 62836087/1029659400, 35819033/617795640, ...

MAPLE

t2:=add( 1/(t1[i]*t1[i+1]*(t1[i]+t1[i+1], )), i=1..nops(t1)-1);

map(numer, ans); # A278047

map(denom, ans); # A278048

CROSSREFS

Cf. A006843, A005728, A240877, A278046.

KEYWORD

nonn,new,frac

STATUS

approved

editing