A238234 Decimal expansion of the alternating sum 1/p(1) - 1/(p(2)*p(3)) + 1/(p(4)*p(5)*p(6)) - 1/(p(7)*p(8)*p(9)*p(10)) + ..., where p(n) is the n-th prime.
4, 3, 4, 3, 2, 7, 7, 0, 3, 1, 9, 6, 9, 3, 8, 1, 0, 2, 2, 9, 6, 1, 5, 7, 5, 1, 3, 0, 2, 4, 8, 3, 7, 2, 3, 6, 7, 4, 2, 7, 9, 1, 3, 8, 9, 2, 7, 7, 1, 9, 6, 7, 7, 9, 3, 8, 5, 5, 2, 6, 0, 1, 4, 1, 4, 4, 2, 1, 1, 5, 0, 5, 4, 1, 6, 0, 9, 4, 6, 8, 0, 4, 0, 7, 3, 8, 9, 6, 1, 9, 8, 6, 8, 6, 1, 4, 2, 9, 1, 5, 2, 7, 8, 5, 7
Offset: 0
Examples
0.4343277031969381022961575130248372367427913892771967793855...
Programs
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Maple
P:=proc(n) local a, b, i, j, k; a:=0.5; k:=1; for i from 2 by 1 to n do b:=1; for j from k by 1 to k+i-1 do b:=b*1/ithprime(j+1); od; k:=j; a:=evalf(a+b*(-1)^(i-1), 105); od; print(a); end: P(100);
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Mathematica
digits = 105; n0 = 10; dn = 10; t[n_] := n*(n + 1)/2; Clear[p]; p[n_] := p[n] = Sum[(-1)^(k + 1)/Product[Prime[j], {j, t[k] - k + 1, t[k]}], {k, 1, n}] // N[#, digits] &; p[n0]; p[n = n0 + dn]; While[RealDigits[p[n]] != RealDigits[p[n - dn]], Print["n = ", n]; n = n + dn]; RealDigits[p[n], 10, digits] // First (* Jean-François Alcover, Aug 12 2014, adapted from PARI *) -
PARI
default(realprecision, 120); T(n) = n*(n + 1)/2; \\ T(n) = A000217(n). sum(k = 1, 100, (-1.)^(k-1)/prod(j = T(k) - k + 1, T(k), prime(j))) \\ Rick L. Shepherd, Mar 07 2014
Extensions
More terms from and offset corrected by Rick L. Shepherd, Mar 07 2014
Comments