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%I A276494 #29 Feb 26 2020 06:44:05
%S A276494 0,4,3,9,0,8,9,6,3,3,5,1,3,0,8
%N A276494 Decimal expansion of the sum of the alternating series of reciprocals of squares of composite numbers.
%C A276494 From _Jon E. Schoenfield_, Feb 02 2018: (Start)
%C A276494 Let S(j) be the partial sum through the j-th term of the alternating series, i.e., S(j) = -Sum_{m=1..j} (-1)^m/A002808(m)^2. The sequence of real values S(2*i-1) for i >= 1, i.e., of partial sums 1/16, 1/16 - 1/36 + 1/64, 1/16 - 1/36 + 1/64 - 1/81 + 1/100, ... (each of which ends with a positive term) will approach the limit from above, while the sequence of real values S(2*i) for i >= 1, i.e., of partial sums 1/16 - 1/36, 1/16 - 1/36 + 1/64 - 1/81, 1/16 - 1/36 + 1/64 - 1/81 + 1/100 - 1/144, ... (each of which ends with a negative term) will approach the limit from below. Let S'(j) = (S(j-1) + S(j))/2; equivalently, S'(j) = -(Sum_{m=1..j-1} (-1)^m/A002808(m)^2 + (1/2)*(-1)^j/A002808(j)^2), so S'(j) can be viewed as an adjusted version of S(j), adjusted by using only half of the final term of S(j). At large values of j, successive values of S'(j) will fluctuate very little compared to the differences between successive values of S(j), because the averaging of successive values of S(j), which are above the limit at each odd value of j and below the limit by very nearly the same amount at each even value of j, causes the values of S'(j) to trace a path midway between that traced by the S(j) values for odd j and those for even j.
%C A276494 Moreover, similar to the situation at A275712, it can be verified that the values of S'(j) themselves fall into three sharply distinct real-valued subsequences: one that converges toward the limit from above and consists of those values where both j and the j-th composite number (i.e., the square root of the reciprocal of the last term in S(j)) are even; one that converges toward the limit from below and consists of those values where j is odd and the j-th composite number is even; and one that stays very near the middle, converging even more rapidly toward the limit, and consisting of all those values where the j-th composite number is odd (regardless of the parity of j). The values in this last subsequence converge very rapidly; see the table in the Example section, which lists values of S'(c_k) where c_k is the smallest odd composite number > 2^k. (End)
%F A276494 Sum_{k>=1} (-1)^(k+1)/A062312(k+1) = 1/16 - 1/36 + 1/64 - 1/81...
%F A276494 Sum_{k>=1} (-1)^(k+1)/((A002808(k))^2) = 1/(4^2) - 1/(6^2) + 1/(8^2) - 1/(9^2)...
%e A276494 Equals 0.04390896...
%e A276494 From _Jon E. Schoenfield_, Feb 02 2018: (Start)
%e A276494 Let S'(j) = -(Sum_{m=1..j-1} (-1)^m/A002808(m)^2 + (1/2)*(-1)^j/A002808(j)^2) and let c_k be the smallest odd composite > 2^k; S'(c_k) quickly converges to a limit, as illustrated below:
%e A276494 k c_k S'(c_k)
%e A276494 == ======== ===============================
%e A276494 3 9 0.04417438271604938271604938...
%e A276494 4 21 0.04390073853615520282186948...
%e A276494 5 33 0.04390758368090798391978693...
%e A276494 6 65 0.04390888269964319809070094...
%e A276494 7 129 0.04390902395888932501501797...
%e A276494 8 259 0.04390896620540588616012725...
%e A276494 9 513 0.04390896281303069589885533...
%e A276494 10 1025 0.04390896330786777379414334...
%e A276494 11 2049 0.04390896335161701542401577...
%e A276494 12 4097 0.04390896335102793828470954...
%e A276494 13 8193 0.04390896335127457473079624...
%e A276494 14 16385 0.04390896335131185998890588...
%e A276494 15 32769 0.04390896335130880417881285...
%e A276494 16 65541 0.04390896335130852088789156...
%e A276494 17 131073 0.04390896335130852182995244...
%e A276494 18 262145 0.04390896335130852702777625...
%e A276494 19 524289 0.04390896335130852688659318...
%e A276494 20 1048577 0.04390896335130852691520992...
%e A276494 21 2097153 0.04390896335130852691785136...
%e A276494 22 4194305 0.04390896335130852691786707...
%e A276494 23 8388609 0.04390896335130852691787563...
%e A276494 24 16777217 0.04390896335130852691787421...
%e A276494 25 33554433 0.04390896335130852691787435...
%e A276494 ...
%e A276494 Extending this several steps farther, it becomes apparent that the limit is 0.04390896335130852691787434869606... (End)
%o A276494 (Sage)
%o A276494 compositessq = (i for i in NN if i>3 and not i.is_prime())
%o A276494 s = RLF(0); s
%o A276494 RealField(110)(s)
%o A276494 for i in range(0, 50000000): s += (-1)^i / next(compositessq)^2
%o A276494 print(s)
%Y A276494 Cf. A002162 (of natural numbers), A072691 (of natural numbers squared), A078437 (of primes), A242301 (of primes squared), A269229 (of composite numbers), A275110 (of composite numbers with distinct prime factors), A275712 (of nonprime numbers).
%Y A276494 Cf. A275712.
%K A276494 nonn,more,cons
%O A276494 1,2
%A A276494 _Terry D. Grant_, Sep 05 2016
%E A276494 a(9)-a(15) from _Robert Price_, Nov 14 2016