A284522 Worst cases for Hart's one-line factorization (OLF) method with multiplier M = 1, see comments.

6, 85, 259, 527, 1177, 1963, 2881, 6403, 6887, 12319, 23701, 40363, 65473, 93011, 144377, 181429, 273487, 337499, 426347, 557983, 702157, 851927, 1044413, 1295017, 1437599, 1763537, 2211119, 2556751, 2982503, 3553027, 3853327

Offset: 1

Author

Charles R Greathouse IV, Mar 28 2017

Keywords

• nonn

Comments

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.

Examples

			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.
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.
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.
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.
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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..214
• William B. Hart, A one line factoring algorithm, Journal of the Australian Mathematical Society 92 (2012), pp. 61-69.

Crossrefs

Subsequence of A138109.

Programs

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

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