

A255051


a(1)=1, a(n+1) = a(n)/gcd(a(n),n) if this GCD is > 1, else a(n+1) = a(n) + n + 1.


4



1, 3, 6, 2, 1, 7, 14, 2, 1, 11, 22, 2, 1, 15, 30, 2, 1, 19, 38, 2, 1, 23, 46, 2, 1, 27, 54, 2, 1, 31, 62, 2, 1, 35, 70, 2, 1, 39, 78, 2, 1, 43, 86, 2, 1, 47, 94, 2, 1, 51, 102, 2, 1, 55, 110, 2, 1, 59, 118, 2, 1, 63, 126, 2, 1, 67, 134, 2, 1, 71, 142, 2, 1
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OFFSET

1,2


COMMENTS

A somehow "trivial" variant of A133058 and A255140, both of which have very similar definitions, but enter 4periodic loops only at later indices.
There could be two motivated values for an initial term: either a(0)=0 which would yield a(1)=1 and the following values via the recursion formula, or a(0)=2 according to the general formula for a(4k).


LINKS

Table of n, a(n) for n=1..73.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,1).


FORMULA

a(4k+1) = 1, a(4k+2) = 4k+3, a(4k+3) = 2*a(4k+2) = 8k+6, a(4k) = 2.
G.f.: x*(1 + 3*x + 6*x^2 + 2*x^3  x^4 + x^5 + 2*x^6  2*x^7)/((1  x)^2*(1 + x)^2*(1 + x^2)^2).  Bruno Berselli, Feb 16 2015
a(n) = ( 2*(3 + (1)^n)  (2  3*n + n*(1)^n)*(1  (1)^((n1)*n/2)) )/4.  Bruno Berselli, Feb 16 2015


EXAMPLE

a(2) = a(1)+2 = 3, a(3) = a(2)+3 = 6, a(4) = a(3)/3 = 2, a(5) = a(4)/2 = 1;
a(6) = a(5)+6 = 7, a(7) = a(6)+7 = 14, a(8) = a(7)/7 = 2, a(9) = a(8)/2 = 1; ...


MATHEMATICA

Table[(2 (3 + (1)^n)  (2  3 n + n (1)^n) (1  (1)^((n  1) n/2)))/4, {n, 1, 80}] (* Bruno Berselli, Feb 16 2015 *)
nxt[{n_, a_}]:={n+1, If[GCD[a, n]>1, a/GCD[a, n], a+n+1]}; Transpose[ NestList[ nxt, {1, 1}, 80]][[2]] (* or *) LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, 1}, {1, 3, 6, 2, 1, 7, 14, 2}, 80] (* Harvey P. Dale, Oct 13 2015 *)


PROG

(PARI) (A255051_upto(N)=vector(N, n, if(gcd(N, n1)>1, N\=gcd(N, n1), N+=n)))(99) \\ simplified by M. F. Hasler, Jan 11 2020
(PARI) A255051(n)=if(n%4>1, if(bittest(n, 0), n*2, n+1), 2bittest(n, 0)) \\ M. F. Hasler, Feb 18 2015
(Magma) &cat [[1, 4*n+3, 8*n+6, 2]: n in [0..20]]; // Bruno Berselli, Feb 16 2015


CROSSREFS

Cf. A133058, A255140.
Sequence in context: A016551 A238555 A176034 * A145896 A159963 A120907
Adjacent sequences: A255048 A255049 A255050 * A255052 A255053 A255054


KEYWORD

nonn,easy


AUTHOR

M. F. Hasler, Feb 15 2015


STATUS

approved



