A177186 a(n+1) = a(n) + p, where p is the largest prime dividing a(n) but not a(n-1), or 1 if there is no such prime.

1, 2, 4, 5, 10, 12, 15, 20, 22, 33, 36, 38, 57, 60, 65, 78, 81, 82, 123, 126, 133, 152, 154, 165, 170, 187, 198, 201, 268, 270, 275, 286, 299, 322, 329, 376, 378, 385, 396, 399, 418, 429, 442, 459, 462, 473, 516, 519, 692, 694, 1041, 1044, 1073, 1110, 1115, 1338

Offset: 1

Author

Franklin T. Adams-Watters, May 04 2010

Keywords

• nonn

Comments

Setting a(1) = 1 is arbitrary; since 1 has no prime divisors, we must then have p = 1 => a(2) = 2, without reference to a(0).

There are three cases where p = 1: n=1 (a(n)=1); n=3 (a(n)=4); n=17 (a(n)=81); and no others through n = 10000. Except that there cannot be such cases consecutively, p=1 iff a(n) is a prime power.

Examples

			After a(9)=22, a(10)=33, the prime divisors of a(10) are 3 and 11; 11 divides 22, so p=3, and a(11)=36.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A076271, A060735, A002620.

Programs

Mathematica

p[n1_, n2_] := If[pp = Complement[Transpose[FactorInteger[n2]][[1]],
Transpose[FactorInteger[n1]][[1]]]; pp == {}, 1, Last[pp]]; a[1] = 1; a[2] = 2; a[n_] := a[n] = a[n-1] + p[a[n-2], a[n-1]]; Table[a[n], {n, 56}] (* Jean-François Alcover, Sep 02 2011 *)
nxt[{a_,b_}]:={b,b+Max[1,Complement[FactorInteger[b][[All,1]],FactorInteger[ a] [[All,1]]]]}; NestList[nxt,{1,2},60][[All,1]] (* Harvey P. Dale, Dec 17 2022 *)

PARI

invec(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lastnotin(vi,vx,dft)=forstep(i=#vi,1,-1,if(!invec(vx,vi[i]),return(vi[i])));dft
al(n)=local(r);r=vector(n);r[1]=1;r[2]=2;for(k=3,n,r[k]=r[k-1]+lastnotin(factor(r[k-1])[,1]~,factor(r[k-2])[,1]~,1));r