A053716 - OEIS

%I #47 Oct 04 2024 11:18:33

%S 7,127,1093,5461,19531,55987,137257,299593,597871,1111111,1948717,

%T 3257437,5229043,8108731,12204241,17895697,25646167,36012943,49659541,

%U 67368421,90054427,118778947,154764793,199411801,254313151,321272407,402321277,499738093

%N a(n) = 1111111 in base n.

%C Evaluation of the seventh cyclotomic polynomial at n. - _Joerg Arndt_, Aug 27 2015

%H Vincenzo Librandi, <a href="/A053716/b053716.txt">Table of n, a(n) for n = 1..1000</a>

%H Carlos M. da Fonseca and Anthony G. Shannon, <a href="https://doi.org/10.7546/nntdm.2024.30.3.491-498">A formal operator involving Fermatian numbers</a>, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.

%H <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = n^6+n^5+n^4+n^3+n^2+n+1 = (n^7-1)/(n-1).

%F G.f.: -x*(x^6-6*x^5+57*x^4+232*x^3+351*x^2+78*x+7)/(x-1)^7. - _Colin Barker_, Oct 29 2012

%F E.g.f.: exp(x)*(1 + 6*x + 57*x^2 + 122*x^3 + 76*x^4+ 16*x^5 + x^6) - 1. - _Stefano Spezia_, Oct 03 2024

%e a(3)=1093 because 1111111 base 3=729+243+81+27+9+3+1=121.

%p A053716 := proc(n)

%p     numtheory[cyclotomic](7,n) ;

%p end proc:

%p seq(A053716(n),n=1..20) ; # _R. J. Mathar_, Feb 07 2014

%t Table[FromDigits["1111111",n],{n,1,30}](*or*)Table[n^6+n^5+n^4+n^3+n^2+n+1,{n,1,60}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2012 *)

%t CoefficientList[Series[-(x^6 - 6 x^5 + 57 x^4 + 232 x^3 + 351 x^2 + 78 x + 7)/(x - 1)^7, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 08 2014 *)

%o (Magma) [7] cat [(n^7-1)/(n-1): n in [2..35]]; // _Vincenzo Librandi_, Feb 08 2014

%Y 7th row of the array A055129.

%Y Cf. A104878.

%K nonn,base,easy

%O 1,1

%A _Henry Bottomley_, Mar 23 2000