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A208507 Reordering of A070776 such that the cyclotomic polynomial Phi(A070776, m) is in sorted order for any integer m >= 2. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
LINKS
EXAMPLE
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.
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] &]
CROSSREFS
Sequence in context: A286451 A102510 A206225 * A216153 A100115 A029670
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Feb 27 2012
STATUS
approved

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Last modified February 22 22:43 EST 2024. Contains 370265 sequences. (Running on oeis4.)