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 A284522 Worst cases for Hart's one-line factorization (OLF) method with multiplier M = 1, see comments. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. 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. EXAMPLE 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. PROG (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 CROSSREFS Subsequence of A138109. Sequence in context: A295229 A330849 A245232 * A167252 A290011 A164266 Adjacent sequences:  A284519 A284520 A284521 * A284523 A284524 A284525 KEYWORD nonn AUTHOR Charles R Greathouse IV, Mar 28 2017 STATUS approved

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Last modified January 25 06:19 EST 2022. Contains 350565 sequences. (Running on oeis4.)