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A336505 5-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_5[x]. 4
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 62, 64, 65, 66, 70, 72, 75, 78, 80, 84, 88, 90, 93, 96, 100, 104, 105, 108, 110, 112, 117, 120, 124, 125, 126, 128, 130, 132, 135, 140, 144, 150 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.

A number m is 5-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007736(d) * n_d, where A007736(d) is the multiplicative order of 5 modulo the largest divisor of d not divisible by 5, and 0 <= n_d <= phi(d)/A007736(d).

The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 46, 286, 2179, 16847, 141446, 1223577, ...

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.

Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.

Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.

Lola Thompson, Variations on a question concerning the degrees of divisors of x^n - 1, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.

Eric Weisstein's World of Mathematics, Finite Field.

Wikipedia, Finite field.

MATHEMATICA

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] >  0]; Select[Range[200], ppQ[#, 5] &]

CROSSREFS

Cf. A132739, A007736, A112765.

Cf. A000010, A260653, A336503, A336504.

Sequence in context: A281624 A242441 A064481 * A303704 A067939 A067784

Adjacent sequences:  A336502 A336503 A336504 * A336506 A336507 A336508

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jul 23 2020

STATUS

approved

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Last modified May 19 11:44 EDT 2022. Contains 353833 sequences. (Running on oeis4.)