login
Numbers not divisible by 2, 3 or 5 (A007775) with digital root 4.
1

%I #27 May 11 2019 16:12:05

%S 13,31,49,67,103,121,139,157,193,211,229,247,283,301,319,337,373,391,

%T 409,427,463,481,499,517,553,571,589,607,643,661,679,697,733,751,769,

%U 787,823,841,859,877,913,931,949,967,1003,1021,1039,1057,1093,1111

%N Numbers not divisible by 2, 3 or 5 (A007775) with digital root 4.

%C Numbers == {13, 31, 49, 67} mod 90 with additive sum sequence 13{+18+18+18+36} {repeat ...}. Includes all prime numbers > 5 with digital root 4.

%H Colin Barker, <a href="/A301622/b301622.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F Numbers == {13, 31, 49, 67} mod 90.

%F From _Colin Barker_, Mar 25 2018: (Start)

%F G.f.: x*(13 + 18*x + 18*x^2 + 18*x^3 + 23*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

%F (End)

%t Rest@ CoefficientList[Series[x (13 + 18 x + 18 x^2 + 18 x^3 + 23 x^4)/((1 - x)^2*(1 + x) (1 + x^2)), {x, 0, 50}], x] (* _Michael De Vlieger_, Apr 21 2018 *)

%t LinearRecurrence[{1,0,0,1,-1},{13,31,49,67,103},50] (* _Harvey P. Dale_, May 11 2019 *)

%o (PARI) Vec(x*(13 + 18*x + 18*x^2 + 18*x^3 + 23*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ _Colin Barker_, Mar 25 2018

%o (GAP) Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=4); # _Muniru A Asiru_, Apr 22 2018

%Y Intersection of A007775 and A017209.

%K nonn,base,easy

%O 1,1

%A _Gary Croft_, Mar 24 2018

%E Last term corrected by _Colin Barker_, Mar 25 2018