A208507 Reordering of A070776 such that the cyclotomic polynomial Phi(A070776, m) is in sorted order for any integer m >= 2.

1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 20, 24, 16, 22, 11, 26, 28, 36, 13, 34, 40, 48, 32, 17, 38, 54, 27, 19, 44, 50, 25, 46, 23, 52, 56, 72, 58, 29, 62, 31, 68, 80, 96, 64, 74, 76, 108, 37, 82, 88, 100, 41, 86, 98, 49, 43, 92, 94, 47, 104, 112, 144

Offset: 1

Author

Lei Zhou, Feb 27 2012

Keywords

• nonn
• easy

Comments

When p is an odd prime number and i >= 1, j >= 1, the cyclotomic polynomial

Phi(2^i*p^j, k)

= Phi(2p,k^(2^(i-1)*p^(j-1)))

= Phi(p, -(k^(2^(i-1)*p^(j-1))))

= (111.....1) (p ones) base -(k^(2^(i-1)*p^(j-1)))

Phi(p^j, k)

= Phi(p, k^(p^(j-1)))

= (111.....1) (p ones) base k^(p^(j-1)).

For odd prime p >= 3, the above numbers can be called "Very Generic Repdigit Numbers".

This sequence is a subsequence of A206225.

The Mathematica program is rewritten to be able to generate this sequence to an arbitrary EulerPhi boundary.

Examples

			The first 20 elements of A206225 are 1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 15, 20, 24, 16, 30, 22, 11.
Among these, 15 = 3 * 5 and 30 = 2 * 3 * 5 cannot be written in the form 2^i*p^j and are thus rejected. So the first 18 terms of this sequence are 1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 20, 24, 16, 22, 11.
		

Crossrefs

Cf. A070776, A206225.

Programs

Mathematica

eb = 48; phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; elim =  Max[Table[Max[phiinv[n]], {n, 2, eb, 2}]]; t = Select[Range[elim], (a = FactorInteger[#]; b = Length[a]; ((b == 1) || ((b == 2) && (a[[1]][[1]] == 2))) && (EulerPhi[#] <= eb)) &]; SortBy[t, Cyclotomic[#, 2] &]