A336505 -id:A336505 - cp's OEIS frontend

A336503 2-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_2[x].

1, 2, 3, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 45, 48, 54, 56, 60, 63, 64, 70, 72, 80, 84, 90, 96, 100, 105, 108, 112, 120, 124, 126, 128, 132, 135, 136, 140, 144, 147, 150, 154, 156, 160, 162, 165, 168, 176, 180, 182, 186, 189, 192

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

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 2-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007733(d) * n_d, where A007733(d) is the multiplicative order of 2 modulo the odd part of d, and 0 <= n_d <= phi(d)/A007733(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 6, 34, 243, 1790, 14703, 120276, 1030279, ...

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.

Cf. A000265, A002326, A007733, A007814.

Cf. A000010, A260653, A336504, A336505.

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[#, 2] &]

A336504 3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 26, 27, 30, 32, 36, 39, 40, 42, 44, 45, 48, 52, 54, 56, 60, 63, 64, 66, 72, 78, 80, 81, 84, 88, 90, 96, 99, 100, 104, 105, 108, 112, 117, 120, 126, 128, 130, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

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 3-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007734(d) * n_d, where A007734(d) is the multiplicative order of 3 modulo the largest divisor of d not divisible by 3, and 0 <= n_d <= phi(d)/A007734(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 41, 258, 1881, 15069, 127350, 1080749, ...

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.

Cf. A038502, A007734, A007949.

Cf. A000010, A260653, A336503, A336505.

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[#, 3] &]

A336506 Lambda-practical numbers: numbers that are p-practical for every rational prime p.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 16, 18, 20, 24, 30, 32, 36, 40, 42, 45, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 105, 108, 112, 120, 126, 128, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 234, 240

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

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. See A336503, A336504 and A336505 for examples.
A number m is lambda-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} lambda(d) * n_d, where lambda(d) = A002322(d) is the Carmichael lambda function, and 0 <= n_d <= phi(d)/lambda(d).
A squarefree number is lambda-practical if and only if it is phi-practical (A260653). All phi-practical numbers are lambda-practical, but there are infinitely many lambda-practical numbers that are not phi-practical: 45, 135, 225, 405, 675, ... (A336507).
If N(x) is the number of terms not exceeding, x then there are two positive constants c_1 and c_2 such that c_1 * x/log(x) <= N(x) <= c_2 * x/log(x) for all x >= 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
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.

Disjoint union of A260653 and A336507.
Cf. A336503, A336504, A336505.
Cf. A000010, A002322, A005117.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; lpQ[n_] := Module[{d = Divisors[n], lam, ns, r, x}, lam = CarmichaelLambda[d]; ns = EulerPhi[d]/lam; r = rep[lam, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] > 0]; Select[Range[250], lpQ]

cp's OEIS Frontend

A336503 2-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_2[x].

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A336504 3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].

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Mathematica

A336506 Lambda-practical numbers: numbers that are p-practical for every rational prime p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica