A248667 Numbers k for which coefficients of the polynomial p(k,x) defined in Comments are relatively prime.

1, 3, 7, 9, 11, 17, 19, 21, 23, 27, 29, 31, 33, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 93, 97, 99, 101, 103, 107, 109, 113, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 147, 149, 151, 153, 157, 159, 161, 163, 167

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• easy

Comments

The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1). For more, see A248664.

Since p(n,x) is a sum of products of terms (n*x + i), the only coefficient which is not necessarily divisible by n is the coefficient of x^0 = A000522(n-1). On the other hand, the coefficient of x^(n-1) is n^n. Therefore n is in this sequence iff gcd(n, A000522(n-1)) = 1. - Peter J. Taylor, Apr 08 2022

From Mikhail Kurkov, Apr 09 2022: (Start)

False conjecture (which still gives many correct values): {b(n)} is a subsequence of {a(n)} where {b(n)} are the numbers m for which Sum(abs(Moebius(p_j+1))) = 0 with m = Product(p_j^k_j). This conjecture was disproved by Peter J. Taylor. The first counterexample, i.e., the smallest m which belongs to {b(n)} and does not belong to {a(n)}, is m = 463. All other counterexamples computed up to 2.5*10^4 have the form 463*b(n). Are there any other numbers q such that q and q*b(n) are counterexamples for any n > 0? [verification needed]

Conjecture: any composite a(n) can be represented as a product a(i)*a(j) (i > 1, j > 1) in at least one way. (End)

Examples

			The first six polynomials with GCD(coefficients) shown just to the right of "=":
p(1,x) = 1
p(2,x) = 2*(x + 1)
p(3,x) = 1*(9x^2 + 12 x +  5)
p(4,x) = 4*(16 x^3 + 28 x^2 + 17 x + 4)
p(5,x) = 5*(125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
p(6,x) = 2*(3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that a(1) = 1 and a(2) = 3.
		

Crossrefs

Cf. A000522, A248664, A248665, A248666, A248668, A248669.

Programs

Mathematica

t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)
u = Table[Apply[GCD, c[n]], {n, 1, 60}]  (* A248666 *)
Flatten[Position[u, 1]]  (* this sequence *)
Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)