A068626 a(3n) = a(3n-1) = 3*n^2, a(3n-2) = 3*n^2 - 3*n + 1.

1, 3, 3, 7, 12, 12, 19, 27, 27, 37, 48, 48, 61, 75, 75, 91, 108, 108, 127, 147, 147, 169, 192, 192, 217, 243, 243, 271, 300, 300, 331, 363, 363, 397, 432, 432, 469, 507, 507, 547, 588, 588, 631, 675, 675, 721, 768, 768, 817, 867, 867, 919, 972, 972, 1027, 1083, 1083, 1141

Offset: 1

Author

Amarnath Murthy, Feb 26 2002

Keywords

• easy
• nonn

Comments

Or, a(1) = 1, a(n) = n + a(n-1) if n does not divide a(n-1) else a(n) = a(n-1). E.g. a(6) = a(5) = 12 as 6 divides 12. a(10) = 10+a(9) = 10+27 = 37.

Links

• B. D. Swan, Table of n, a(n) for n = 1..90000
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A091684 (first differences).

Programs

Magma

[(n mod 3 eq 1) select (n+2)^2/3 - n-1 else (n+((n mod 3)^2) mod 3 )^2/3  : n in [1..50]]; // Marius A. Burtea, Feb 19 2020

Mathematica

LinearRecurrence[{1,0,2,-2,0,-1,1},{1,3,3,7,12,12,19},60] (* Harvey P. Dale, Jun 29 2022 *)

PARI

Vec(x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^50)) \\ Colin Barker, Feb 19 2020

Perl

my @a = (1); for (my $n = 1; $n <= 90000; $n ++) {
  $a[$n] = $a[$n - 1] + ($a[$n - 1] % $n != 0 ? $n : 0);
  print "$n $a[$n]\n";
} # Georg Fischer Feb 18 2020

Formula

From Colin Barker, Feb 18 2020: (Start)

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

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>7.

(End)

Sum_{n>=1} 1/a(n) = Pi/sqrt(3)*tanh(Pi/(2*sqrt(3))) + Pi^2/9. - Amiram Eldar, Sep 21 2023

Extensions

Entry revised by N. J. A. Sloane, Mar 13 2006