|
%I #54 Mar 13 2023 06:52:45
%S 0,1,19,343,6175,111151,2000719,36012943,648232975,11668193551,
%T 210027483919,3780494710543,68048904789775,1224880286215951,
%U 22047845151887119,396861212733968143,7143501829211426575,128583032925805678351
%N a(n) = (18^n-1)/17.
%C Partial sums of powers of 18 (A001027), q-integers for q=18: diagonal k=1 in triangle A022182.
%C Partial sums are in A014901. Also, the sequence is related to A014935 by A014935(n) = n*a(n) - Sum_{i=0..n-1} a(i), for n>0. - _Bruno Berselli_, Nov 06 2012
%C From _Bernard Schott_, May 06 2017: (Start)
%C Except for 0, 1 and 19, all terms are Brazilian repunits numbers in base 18, and so belong to A125134. From n = 3 to n = 8286, all terms are composite. See link "Generalized repunit primes".
%C As explained in the extensions of A128164, a(25667) = (18^25667 - 1)/17 would be (is) the smallest prime in base 18. (End)
%H Vincenzo Librandi, <a href="/A218721/b218721.txt">Table of n, a(n) for n = 0..800</a>
%H Harvey Dubner, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185243-9">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930.
%H <a href="/index/Par#partial">Index entries related to partial sums</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (19,-18).
%F a(n) = floor(18^n/17).
%F G.f.: x/((1-x)*(1-18*x)). - _Bruno Berselli_, Nov 06 2012
%F a(n) = 19*a(n-1) - 18*a(n-2). - _Vincenzo Librandi_, Nov 07 2012
%F E.g.f.: exp(x)*(exp(17*x) - 1)/17. - _Stefano Spezia_, Mar 11 2023
%e a(3) = (18^3 - 1)/17 = 343 = 7 * 49; a(6) = (18^6 - 1)/17 = 2000719 = 931 * 2149. - _Bernard Schott_, May 01 2017
%t LinearRecurrence[{19, -18}, {0, 1}, 40] (* _Vincenzo Librandi_, Nov 07 2012 *)
%t Join[{0},Accumulate[18^Range[0,20]]] (* _Harvey P. Dale_, Nov 08 2012 *)
%o (PARI) A218721(n)=18^n\17
%o (Maxima) A218721(n):=(18^n-1)/17$ makelist(A218721(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (Magma) [n le 2 select n-1 else 19*Self(n-1)-18*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 07 2012
%Y Cf. A000225, A001027, A002275, A002450, A002452, A003462, A003463, A003464, A014901, A014935, A016123, A016125, A022182, A023000, A023001, A064108, A091030, A091045, A094028, A125134, A128164, A131865, A135518, A135519, A218722, A218724, A218733, A218743, A218752.
%K nonn,easy
%O 0,3
%A _M. F. Hasler_, Nov 04 2012
|
