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A248667
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Numbers k for which coefficients of the polynomial p(k,x) defined in Comments are relatively prime.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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
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)
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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