A301701 a(n) is the smallest positive integer m, with the property that n appears as a coefficient in the polynomial P_m(x) = (x-1)(x^2-1)...(x^m-1).

3, 1, 4, 10, 12, 17, 16, 19, 20, 22, 22, 23, 24, 25, 25, 25, 24, 26, 26, 28, 27, 27, 29, 28, 28, 29, 29, 30, 28, 29, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31, 31, 32, 33, 33, 33, 32, 33, 32, 33, 32, 33, 33, 36, 35, 33, 33, 36, 34, 34, 37, 35, 34, 37, 35, 34, 34, 35, 35, 35, 35

Offset: 0

Author

Ovidiu Bagdasar, Mar 25 2018

Keywords

• nonn

Comments

We conjecture that all integers appear as a coefficient of a polynomial P_m(x).

This property is known to hold for the cyclotomic polynomials.

The conjecture holds for the first 10^5 positive integers, with a maximum on those integers of a(99852) = 1921. - David A. Corneth, Apr 08 2018

Examples

			We have:
P_1(x) = x-1, hence a(1)=1.
P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1;
P_3(x) = (x-1)*(x^2-1)*(x^3-1) = x^6-x^5-x^4+x^2+x-1;
P_4(x) = (x-1)*(x^2-1)*(x^3-1)*(x^4-1) = x^10 - x^9 - x^8+2x^5-x^2-x+1, hence a(2)=4.
n=3 first appears as a coefficient of P_{10}(x).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
• Jiro Suzuki, On coefficients of cyclotomic polynomials, Proc. Japan Acad. Ser. A Math. Sci. 63:7 (1987), pp. 279-280.

Crossrefs

Cf. A231599: a(n) is the index of the first row m containing number n.

Programs

Maple

T:= proc(n) option remember; [(p-> seq(coeff(p, x, i),
    i=0..degree(p)))(expand(mul(1-x^i, i=1..n)))] end:
a:= proc(n) local k; for k while not n in T(k) do od: k end:
seq(a(n), n=0..70);  # Alois P. Heinz, Mar 29 2019

Mathematica

With[{s = Array[CoefficientList[Times @@ Array[x^# - 1 &, #], x] &, 40]}, TakeWhile[Array[FirstPosition[s, #][[1]] &, Max@ Map[Max, s]], IntegerQ]] (* Michael De Vlieger, Apr 05 2018 *)

PARI

a(n) = {my(k=1); while (!vecsearch(vecsort(Vec(prod(j=1, k, x^j-1))), n), k++); k;} \\ Michel Marcus, Apr 08 2018

PARI

first(n) = {my(pol = [1], res = vector(n), todo = n+1, t = 0); while(1, t++; for(i = 1, #pol, if(0 < pol[i] && pol[i] <=n, if(res[pol[i]] == 0, res[pol[i]] = t-1; todo--; if(todo == 0, return(concat([3], res))))));  pol = concat(pol, vector(t)) - concat(vector(t), pol))} \\ David A. Corneth, Apr 08 2018

Extensions

Offset changed to 0 by David A. Corneth, Apr 08 2018