A284522 - OEIS

%I #11 Apr 05 2017 16:36:13

%S 6,85,259,527,1177,1963,2881,6403,6887,12319,23701,40363,65473,93011,

%T 144377,181429,273487,337499,426347,557983,702157,851927,1044413,

%U 1295017,1437599,1763537,2211119,2556751,2982503,3553027,3853327

%N Worst cases for Hart's one-line factorization (OLF) method with multiplier M = 1, see comments.

%C Hart's algorithm begins with trial division to the cube root of the number and a check for squares, so numbers factored by these means are removed (leaving A138109). The remaining numbers are compared on the basis of the number of steps Hart's algorithm requires to factor them; new records are members of this sequence.

%H Charles R Greathouse IV, <a href="/A284522/b284522.txt">Table of n, a(n) for n = 1..214</a>

%H William B. Hart, <a href="http://wrap.warwick.ac.uk/54707/1/WRAP_Hart_S1446788712000146a.pdf">A one line factoring algorithm</a>, Journal of the Australian Mathematical Society 92 (2012), pp. 61-69.

%e OLF factors 6 on step 2: s = ceil(sqrt(2*6)) = 4, s^2 = 4 mod 6; 4 = 2^2, gcd(6, 4-2) = 2.

%e OLF factors 85 on step 3: s = ceil(sqrt(3*85)) = 16, s^2 = 1 mod 85; 1 = 1^2, gcd(85, 16-1) = 5.

%e OLF factors 259 on step 5: s = ceil(sqrt(5*259)) = 36, s^2 = 1 mod 259; 1 = 1^2, gcd(259, 36-1) = 7.

%e OLF factors 527 on step 8: s = ceil(sqrt(8*527)) = 65, s^2 = 9 mod 527; 9 = 3^2, gcd(527, 65-3) = 31.

%e OLF factors 1177 on step 9: s = ceil(sqrt(9*1177)) = 103, s^2 = 16 mod 1177; 16 = 4^2, gcd(1177, 103-4) = 11.

%o (PARI) listA138109(lim)=if(lim<6, return([])); my(v=List([6])); forprime(p=3, sqrtint(1+lim\=1)-1, forprime(q=p+2, min(p^2-2, lim\p), listput(v, p*q))); Set(v)

%o g(n)=for(i=1, n, if(issquare((sqrtint(i*n-1)+1)^2%n), return(i)))

%o list(lim)=my(u=Vecsmall(listA138109(lim)),v=List(),r,t); for(i=1,#u, t=g(u[i]); if(t>r, r=t; listput(v,u[i]))); u=0; Vec(v) \\ _Charles R Greathouse IV_, Mar 28 2017

%o (PARI) make(from,to)=my(v=List()); from=ceil(from); forprime(p=max(sqrtnint(from,3)+1,3),sqrtint(1+to\=1)-1, forprime(q=max(p+2,from/p),min(p^2-2,to\p), listput(v,p*q))); Set(v)

%o g(n)=for(i=1, n, if(issquare((sqrtint(i*n-1)+1)^2%n), return(i)))

%o list(lim)=my(u,v=List([6]),r,t,step=10^7); forstep(n=85,lim,step, u=make(n,min(n+step-1,lim)); for(i=1,#u,t=g(u[i]); if(t>r, r=t; listput(v,u[i]); print1(u[i]", ")))); Vec(v) \\ _Charles R Greathouse IV_, Apr 03 2017

%Y Subsequence of A138109.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, Mar 28 2017