A301622 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 4.

13, 31, 49, 67, 103, 121, 139, 157, 193, 211, 229, 247, 283, 301, 319, 337, 373, 391, 409, 427, 463, 481, 499, 517, 553, 571, 589, 607, 643, 661, 679, 697, 733, 751, 769, 787, 823, 841, 859, 877, 913, 931, 949, 967, 1003, 1021, 1039, 1057, 1093, 1111

Offset: 1

Author

Gary Croft, Mar 24 2018

Keywords

• nonn
• base
• easy

Comments

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.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Intersection of A007775 and A017209.

Programs

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

Mathematica

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 *)
LinearRecurrence[{1,0,0,1,-1},{13,31,49,67,103},50] (* Harvey P. Dale, May 11 2019 *)

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

Formula

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

From Colin Barker, Mar 25 2018: (Start)

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

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

(End)

Extensions

Last term corrected by Colin Barker, Mar 25 2018