A218721 a(n) = (18^n-1)/17.

0, 1, 19, 343, 6175, 111151, 2000719, 36012943, 648232975, 11668193551, 210027483919, 3780494710543, 68048904789775, 1224880286215951, 22047845151887119, 396861212733968143, 7143501829211426575, 128583032925805678351

Offset

0,3

Comments

Partial sums of powers of 18 (A001027), q-integers for q=18: diagonal k=1 in triangle A022182.

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

From Bernard Schott, May 06 2017: (Start)

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".

As explained in the extensions of A128164, a(25667) = (18^25667 - 1)/17 would be (is) the smallest prime in base 18. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.

Index entries related to partial sums

Index entries for linear recurrences with constant coefficients, signature (19,-18).

Formula

a(n) = floor(18^n/17).

G.f.: x/((1-x)*(1-18*x)). - Bruno Berselli, Nov 06 2012

a(n) = 19*a(n-1) - 18*a(n-2). - Vincenzo Librandi, Nov 07 2012

E.g.f.: exp(x)*(exp(17*x) - 1)/17. - Stefano Spezia, Mar 11 2023

Example

a(3) = (18^3 - 1)/17 = 343 = 7 * 49; a(6) = (18^6 - 1)/17 = 2000719 = 931 * 2149. - Bernard Schott, May 01 2017

Mathematica

LinearRecurrence[{19, -18}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0}, Accumulate[18^Range[0, 20]]] (* Harvey P. Dale, Nov 08 2012 *)

Prog

(PARI) A218721(n)=18^n\17
(Maxima) A218721(n):=(18^n-1)/17$ makelist(A218721(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(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

Crossrefs

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.

Sequence in context: A170652 A170700 A170738 * A282030 A192568 A171324

Adjacent sequences: A218718 A218719 A218720 * A218722 A218723 A218724

Keyword

nonn,easy

Author

M. F. Hasler, Nov 04 2012

Status

approved