A066730 Numbers with ever-increasing minimal-square-deniers.

2, 3, 12, 21, 60, 184, 280, 364, 1456, 3124, 5236, 17185, 25249, 49504, 233776, 364144, 775369, 3864169, 8794864, 10869664, 32384209, 105361344, 173899609, 425088976, 2140195264, 2805544681, 10310263441, 11940537961, 38432362081, 43395268849, 51802119889, 299530084681

Offset: 1

Author

Phil Carmody, Jan 15 2002

Keywords

• nonn

Comments

The Jacobi of modular reductions of a number is often used by a bignum library to give a quick (negative) answer to the question of whether an integer is an exact square. This sequence gives the cutoffs for ever-increasing numbers of required modular tests, on the assumption that one is avoiding a brute force square-root/square/compare. All terms to 8794864 found by Jack Brennen.

Examples

			2 is 'square-denied' by 3, as 2 is not a quadratic residue mod 3 3 is square-denied by 2^2=4, but not by any lower prime power (2 or 3) 12 has 5 as its minimal square-denier (0 mod 2, 0 mod 3, 0 mod 4 all QRs) 21 has 2^3=8 as its minimal square-denier. (note that 24 has 7 as its minimal square-denier, the first number with that property, but it is larger than 21)
		

Links

• J. Brennen, discussion about issquare() tests without use of sqrt() on Caldwell's 'primenumbers' list.
• Jack Brennen and others, Programming Question - detecting squareness, digest of 32 messages in primenumbers Yahoo group, in particular message 8, Jan 15 - Jan 22, 2002.

Programs

Mathematica

mo = Select[Range[3, 60], Length[FactorInteger[#]] == 1 &]; T = Table[0 Range[mo[[i]]], {i, Length@ mo}]; Do[T[[i, 1 + Mod[j^2, mo[[i]]]]] = 1, {i, Length@mo}, {j, mo[[i]]}]; w[n_] := If[IntegerQ@ Sqrt@ n, -1, Block[{k=1}, While[k < Length[mo] && T[[k, 1 + Mod[n, mo[[k]]]]] == 1, k++]; k-1]]; rec = -1; n = 1; L = {}; While[n < 8 10^5, n++; v = w[n]; If[v > rec, rec = v; AppendTo[L, n]]]; L (* computes first 17 terms, Giovanni Resta, Nov 15 2019 *)

Extensions

Offset set to 1 and terms a(20)-a(32) added by Giovanni Resta, Nov 15 2019