A118031 Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.
3, 3, 7, 0, 2, 8, 3, 2, 5, 9, 4, 9, 7, 3, 7, 3, 3, 2, 0, 4, 9, 2, 1, 5, 7, 2, 9, 8, 5, 0, 5, 5, 3, 1, 1, 2, 3, 0, 7, 1, 4, 5, 7, 7, 7, 9, 4, 5, 2, 7, 7, 8, 4, 9, 1, 3, 3, 5, 0, 6, 8, 9, 2, 5, 9, 8, 2, 5, 1, 9, 7, 6, 0, 3, 4, 9, 4, 7, 6, 7, 5, 8, 9, 7, 0, 3, 0, 1
Offset: 1
Examples
3.3702832594973733204921572985...
Links
- Joseph Myers, Table of n, a(n) for n = 1..1001
- Joseph Myers, Polynomial-time algorithm.
- Michael Penn, Does this series converge??, YouTube video, 2021.
- Radovan Potůček, Formulas for the Sums of the Series of Reciprocals of the Polynomial of Degree Two with Non-zero Integer Roots, Algorithms as a Basis of Modern Applied Mathematics, Studies in Fuzziness and Soft Computing book series (STUDFUZZ, Vol. 404) Springer (2021), 363-382.
- Eric Weisstein's World of Mathematics, Palindromic Number.
Programs
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Mathematica
NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits@ n}, If[ Union@ idn == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[ idn, Ceiling[l/2]]]] > FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[Join[Take[idn, Ceiling[l/2]], Reverse[Take[idn, Floor[l/2]]]]], idfhn = FromDigits[Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits@ idfhn, Drop[ Reverse[ IntegerDigits@ idfhn], Mod[l, 2]]]]]]]]; pal = 1; sm = 0; Do[ While[pal < 10^n + 1, sm = N[sm + 1/pal, 128]; pal = NextPalindrome@ pal]; Print[{n, sm}], {n, 0, 17}] (* Robert G. Wilson v, Oct 20 2010 *)
Formula
a(n) = Sum_{palindromes p>0} 1/p.
a(n) = Sum_{n>=2} 1/A002113(n).
Extensions
Corrected by Eric W. Weisstein, May 14 2006
Corrected and extended by Robert G. Wilson v, Oct 20 2010
Corrected and extended by Joseph Myers, Jun 26 2014
Comments