cp's OEIS Frontend

A295838 Largest value corresponding to a string of n printable ASCII characters.

%I A295838 #51 Sep 08 2022 08:46:20
%S A295838 0,126,32382,8289918,2122219134,543288098430,139081753198206,
%T A295838 35604928818740862,9114861777597660798,2333404615065001164414,
%U A295838 597351581456640298090110
%N A295838 Largest value corresponding to a string of n printable ASCII characters.
%C A295838 The n-th term of this sequence is the result of forming an ASCII (American Standard Code for Information Exchange) text string of n characters using the (printable) character with the largest binary value and then converting the binary value of the string to base 10. a(n) is therefore a measure of the largest possible size of an ASCII string with n printable characters. This sequence uses standard 7-bit ASCII; A175824 is the same sequence using 8-bit Extended ASCII.
%C A295838 Conjecture: For every a(n) there exists a sequence of primes (p(1), p(2), p(3), ...)  such that for each term a(n) there exists a set of primes that when added to the term result in another prime. For example, a(2)=126 and 126 + {5,11,13,23,37,41,47,...} all are primes.
%C A295838 Corollary 1: If it is the case that the size of the set of prime numbers is countably infinite, then the cardinality of the sequence that contains the sequence of primes p(1), p(2), ... that produce a new prime for every a(n) is uncountably infinite ... [ (a(1)+p(1), a(1)+p(2), a(1)+p(3), ...), (a(2)+p(1)', a(2)+p(2)', a(2)+p(3)'), ...)
%H A295838 Iain Fox, <a href="/A295838/b295838.txt">Table of n, a(n) for n = 0..415</a>
%H A295838 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (257,-256).
%F A295838 a(n) = (42/85)*(256^n - 1).
%F A295838 From _Iain Fox_, Nov 28 2017: (Start)
%F A295838 G.f.: 126*x/((1-256*x)*(1-x)).
%F A295838 E.g.f.: 42/85*(e^(256*x)-e^x).
%F A295838 a(n) = 42/85 * A175824(n).
%F A295838 (End)
%F A295838 For n > 0, a(n) = 256*a(n-1) + 126. - _Jon E. Schoenfield_, Nov 29 2017
%F A295838 For n > 1, a(n) = 257*a(n-1) - 256*a(n-2). - _Iain Fox_, Jan 02 2018
%e A295838 The lexicographically last 2-character printable ASCII string is "~~", which is 7E7E in hexadecimal or 32382 in decimal, thus a(2) = 32382.
%p A295838 A295838:=n->(42/85)*(256^n - 1): seq(A295838(n), n=0..20); # _Wesley Ivan Hurt_, Nov 29 2017
%t A295838 Array[(42/85) (256^# - 1) &, 11, 0] (* _Michael De Vlieger_, Dec 11 2017 *)
%t A295838 CoefficientList[Series[126 x/((1 - 256 x) (1 - x)), {x, 0, 10}], x] (* _Michael De Vlieger_, Dec 11 2017 *)
%o A295838 (PARI) a(n) = 42/85*(256^n-1) \\ _Iain Fox_, Nov 28 2017
%o A295838 (PARI) first(n) = Vec(126*x/((1-256*x)*(1-x)) + O(x^n), -n) \\ _Iain Fox_, Nov 28 2017
%o A295838 (Magma) [(42/85)*(256^n - 1) : n in [0..20]]; // _Wesley Ivan Hurt_, Nov 29 2017
%Y A295838 Cf. A175824.
%K A295838 nonn,easy,base
%O A295838 0,2
%A A295838 _Abilene Sukin_, Nov 28 2017