A301617 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 1.

1, 19, 37, 73, 91, 109, 127, 163, 181, 199, 217, 253, 271, 289, 307, 343, 361, 379, 397, 433, 451, 469, 487, 523, 541, 559, 577, 613, 631, 649, 667, 703, 721, 739, 757, 793, 811, 829, 847, 883, 901, 919, 937, 973, 991, 1009, 1027, 1063, 1081, 1099

Offset: 1

Author

Gary Croft, Mar 24 2018

Keywords

• nonn
• base
• easy

Comments

Numbers == {1, 19, 37, 73} mod 90 with additive sum sequence 1{+18+18+36+18} {repeat ...}. Includes all prime numbers > 7 with digital root 1.

Examples

			1+18=19; 19+18=37; 37+36=73; 73+18=91; 91+18=109.
		

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

Cf. A177704, A045572.

Intersection of A007775 and A017173.

Programs

Maple

seq(seq(i+90*j,i=[1,19,37,73]),j=0..30); # Robert Israel, Mar 25 2018

Mathematica

LinearRecurrence[{1,0,0,1,-1},{1,19,37,73,91},50] (* Harvey P. Dale, Dec 14 2019 *)

PARI

a(n) = 1 + 18 * (n - 1 + n\4) \\ David A. Corneth, Mar 24 2018

PARI

Vec(x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 24 2018

Formula

n == {1, 19, 37, 73} mod 90.

a(n + 1) = a(n) + 18 * A177704(n + 1). - David A. Corneth, Mar 24 2018

From Colin Barker, Mar 24 2018: (Start)

G.f.: x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*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

The missing term 1081 added to the sequence by Colin Barker, Mar 24 2018