A206284 -id:A206284 - cp's OEIS frontend

A208181 Numbers that match polynomials over {0,1} that have a factor containing 3 as a coefficient; see Comments.

2229, 2613, 2757, 2769, 4458, 5226, 5514, 5538, 7335, 8373, 8421, 8589, 8853, 8913, 8916, 8919, 8949, 9093, 9485, 10293, 10311, 10353, 10389, 10437, 10452, 10461, 10563, 10677, 10689, 10821, 10833, 10839, 10869, 11013, 11028, 11031

Offset: 1

Clark Kimberling, Feb 24 2012

nonn

The polynomials having coefficients in {0,1} are enumerated at A206073.  They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
p(2229,x) =1 + x^2 + x^4 + x^5 + x^7 + x^11= (1+x)*f(x), where f(x) =  1 - x + 2 x^2 - 2 x^3 + 3 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8 - x^9 + x^10.  This show that a factor of p(2229,x) has a factor that has 3 as a coefficient.  Actually, 2229 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2}.
The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073.  The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208181 are disjoint.

Cf. A208179, A206073, A206284, A208180, A208182.

t = Table[IntegerDigits[n, 2], {n, 1, 15000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 15000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, 3, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 14900}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208181 *)

A208182 Numbers that match polynomials over {0,1} that have a factor containing -3 as a coefficient; see Comments.

Original entry on oeis.org

8421, 8853, 9093, 10311, 10353, 10389, 10437, 10563, 10689, 10821, 10833, 10839, 10869, 11157, 12183, 12453, 14469, 14973, 14997, 16779, 16842, 17055, 17465, 17706, 18186, 18515, 18639, 19985, 20025, 20622, 20643, 20706, 20778

Offset: 1

Clark Kimberling, Feb 24 2012

nonn

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
p(8421,x) = 1 + x^2 + x^5 + x^6 + x^7 + x^13
= (1 + x)*(1 + x + x^2)*f(x), where
f(x) = 1 - 2*x + 3*x^2 - 3*x^3 + 2*x^4 - x^7 + 2*x^8 - 2*x^9 + x^10.
This show that a factor of p(8421,x) has a factor that has -3 as a coefficient. Actually, 8421 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2,3}.
The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208181 are disjoint.

Cf. A208179, A206073, A206284, A208180, A208181.

t = Table[IntegerDigits[n, 2], {n, 1, 25000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 25000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, -3, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 24900}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208182 *)
(* Peter J. C. Moses, Feb 22 2012 *)

A167219 Numbers k such that there exists a positive integer B for which k = Sum_{i=0..m} (B^i)*a_i where the a_i are defined by k = Product_{i=0..m} prime(i+1)^a_i.

Original entry on oeis.org

3, 6, 10, 12, 24, 27, 36, 48, 96, 100, 144, 175, 192, 216, 273, 384, 486, 576, 768, 972, 1296, 1536, 1728, 2304, 3072, 3125, 6144, 9216, 12288, 13824, 17496, 19683, 20736, 24576, 36864, 46656, 49152, 62208, 69984, 98304, 110592, 147456, 196608, 331776, 393216, 589824

Offset: 1

Ctibor O. Zizka, Oct 30 2009

nonn

Previous name: Numbers k such that there exists a solution to (p_m ^ a_m)*(p_m-1 ^ a_m-1)*...*(3^a_1)*(2^a_0) = (B^m)*a_m + (B^m-1)*a_m-1 + ... + (B^1)*a_1 + (B^0)*a_0 where k = (p_m ^ a_m)*(p_m-1 ^ a_m-1)*...*(3^a_1)*(2^a_0); a_m >= 1; a_(i= 0; p_0, p_1, ..., p_m are prime numbers; a_0, a_1, ..., a_m, B are integers.
B is the base in which we can express k as Sum_{i=0..m} B^i * a_i. B may also be seen as the variable in a polynomial, and k is then also an encoding of the polynomial (defined by the product of primes formula).
For k = (2^r)*3 we have B = (2^r)*3 - r.
A167221(n) is the smallest positive integer that yields a solution for k = a(n).
Negative B's can be obtained when the polynomial is an even function. This happens for instance when for k = 10, 100, 3125, ... - Michel Marcus, Aug 10 2022
From Peter Munn, Aug 13 2022: (Start)
Positive integers k such that k is a fixed point of a completely additive function f_B:N+ -> Z, B > 0, where f_B(prime(i+1)) = B^i for all i >= 0. Equivalently, since row B of A104244 is f_B, {a(n)} lists the columns of A104244 that contain their own column number.
If we require B to be negative instead, the sequence appears to start 10, 100, 3125, 1799875, 65610000, ... . Of these, 1799875 = 5^3 * 7 * 11^2 * 17 is the only k with only negative solutions (B = -11); the solutions for 65610000 are {4049, -4051}.
(End)
If p is the (k+1)-th prime and p is congruent to 1 modulo k, then p^p is a term with p^((p-1)/k) a solution for B. The list of such primes starts 3, 5, 7, 31, 97, 101, 331, ... . I suspect this list is infinite, meaning the greatest prime factor of the terms would be unbounded. - Peter Munn, Aug 15 2022

			For k = 10 = 2^1 * 3^0 * 5^1, k = B^0 * 1 + B^1 * 0 + B^2 * 1, so we have to solve the equation 10 = 1 + B^2 for a positive integer B, B = 3. But B=-3 works too. Thus 10 is a term.
For k = 12 = 2^2 * 3^1, k = B^0 * 2 + B^1 * 1, so we have to solve the equation 12 = 2 + B for a positive integer B. B = 10. Thus 12 is a term.
For k = 21 = 2^0 * 3^1 * 5^0 * 7^1, k = B^0 * 0 + B^1 * 1 + B^2 * 0 + B^3 * 1, so we have to solve the equation 21 = B + B^3 for an integer B. No such B exists, so 21 is not a term of this sequence.
From _Michel Marcus_, Aug 10 2022: (Start)
In other words:
  10 is a term because 10 = 5^1 * 3^0 * 2^1 and 101 in base 3 is 10.
  12 is a term because 12 = 3^1 * 2^2 and 12 in base 10 is 12. (End)

David A. Corneth, Table of n, a(n) for n = 1..98 (terms <= 10^10)
David A. Corneth, PARI program

Cf. A054841, A054842, A067255, A104244, A167221.

A206284 describes the polynomial encoding used here.

isok(k) = if (k>1, my(f=factor(k), v=primes(primepi(vecmax(f[,1])))); my(p=sum(i=1, #v, 'x^(i-1)*valuation(k,v[i]))); p -= k; my(c=-polcoef(p, 0)); my(q=(p+c)/x); my(d=divisors(c)); for (k=1, #d, if(subst(q, x, d[k]) == c/d[k], return(1)););); \\ Michel Marcus, Aug 08 2022

\\ See PARI link \\ David A. Corneth, Aug 10 2022

from sympy import divisors, factorint, sieve
def ok(n):
    if n < 2: return False
    f = factorint(n)
    a = [f[pi] if pi in f else 0 for pi in sieve.primerange(2, max(f)+1)]
    for B in range(1, n+1):
        polyB = sum(B**i*ai for i, ai in enumerate(a) if ai > 0)
        if polyB == n: return True
        elif polyB > n: return False
    return False
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Aug 10 2022

Edited by Jon E. Schoenfield, Mar 16 2022
Incorrect term 71 removed, new name and more terms from Michel Marcus, Aug 08 2022
a(41)-a(46) from Michael S. Branicky, Aug 10 2022

A206285 Numbers that match polynomials not irreducible over the nonnegative integers.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 26, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 69, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 99, 101, 103

Offset: 1

Clark Kimberling, Feb 05 2012

nonn

Complement of A206284.

			(See the example at A206284.)

Cf. A206285, A206331.

b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 400;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x], {n, 1, z/4}]
v = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[v, n]], {n, z/2}]; v  (* A206284 *)
Complement[Range[200], v]      (* A206285 *)

A206330 Numbers that match polynomials irreducible over the integers.

Original entry on oeis.org

3, 4, 5, 6, 9, 10, 17, 18, 19, 20, 21, 22, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 53, 54, 55, 56, 57, 58, 59, 60, 69, 70, 73, 74, 77, 78, 81, 82, 83, 84, 87, 88, 97, 98, 101, 102, 105, 106, 109, 110, 113, 114, 117, 118, 119, 120, 123

Offset: 1

Clark Kimberling, Feb 06 2012

nonn

Each n>1 matches a polynomial having integer coefficients
determined by the prime factorization of n.  Let c be a
positive integer, and write
c=p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k), and
define p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
If c/d is a rational number with GCD(c,d)=1, define
Q(c/d,x)=p(c,x)-p(d,x).  Let c(n)/d(n) be the n-th
positive rational number given by the canonical
bijection; i.e., c(n)=A038568(n)/A038569(n).
Define P(0,x)=1 and P(n,x)=Q(c(n)/d(n),x).  Polynomials
having nonnegative integer coefficients are matched to
the nonnegative integers as follows:
...
n .... P[n,x] .. irreducible
0 .... 0 ....... no
1 ... -1 ....... no
2 .... 1 ....... no
3 ... -x ....... yes
4 .... x ....... yes
5 ... 1-x ...... yes
6 .. -1+x ...... yes
7 .. -2 ........ no
8 ... 2 ........ no
9 .. -2+x ...... yes
10 .. 2-x ...... yes

			In the table under Comments, read "yes" for n=3,4,5,6,9,10.

Cf. A206284 (polynomials over the positive integers),

A206331 (complement of A206330).

b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 1000;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
 == Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
c[n_] := Module[{s = 1, k = 2, j = 1},
   While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
   s = s - 2*EulerPhi[k - 1];
   While[s <= n, If[GCD[j, k - 1]
      == 1, s = s + 2]; j = j + 1];
   If[s > n + 1, j - 1, k - 1]];
d[n_] := Module[{s = 1, k = 2, j = 1},
   While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
   s = s - 2*EulerPhi[k - 1];
   While[s <= n, If[GCD[j, k - 1]
      == 1, s = s + 2]; j = j + 1];
   If[s > n + 1, k - 1, j - 1]];
P[n_, x_] :=
 u[[c[n]]].b[-1 + Length[u[[c[n]]]]] -
  u[[d[n]]].b[-1 + Length[u[[d[n]]]]]
TableForm[Table[{n, P[n, x], Factor[P[n, x]]},
   {n, 1, z/4}]];
v = {}; Do[n++;
 If[IrreduciblePolynomialQ[P[n, x]], AppendTo[v, n]], {n, z/2}]
v                            (* A206330 *)
Complement[Range[0,200], v]  (* A206331 *)

A208135 Numbers that match polynomials over {0,1} that have a factor containing a negative coefficient.

Original entry on oeis.org

9, 18, 21, 27, 33, 35, 36, 39, 42, 45, 49, 54, 57, 63, 65, 66, 70, 72, 75, 78, 84, 90, 93, 98, 99, 105, 108, 114, 126, 129, 130, 132, 133, 135, 140, 141, 144, 147, 150, 153, 155, 156, 159, 161, 165, 168, 175, 177, 180, 183, 186, 189, 195, 196, 198, 201

Offset: 1

Clark Kimberling, Feb 23 2012

nonn

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
  p(1,x) = 1
  p(2,x) = x
  p(3,x) = x + 1
  p(9,x) = x^3 + 1 = (x + 1)*(x^2 - x + 1)
  p(18,x) = x*(x + 1)*(x^2 - x + 1)
  p(33,x) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).
A208135 gives those n for which p(n,x) has a factor containing a negative coefficient; A208136 is a subsequence of A208135 in which, for each p(n,x), there is a factor containing a negative coefficient, and that factor has not already occurred for some p(k,x) with k

			The first few polynomial factors having a negative coefficient are as follows:
  x^2 - x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
  x^4 - x^3 + x^2 - x + 1 divides p(n,x) for n=33,66,...
  x^3 - x^2 + 1 divides p(n,x) for n=35,70,...
  x^4 - x^3 + x^2 + 1 divides p(n,x) for n=39,...
  x^3 - x + 1 divides p(n,x) for n=49,...
  x^4 + x^2 - x + 1 divides p(n,x) for n=57,...
In A208136, the duplicates (such as 18, 21, 27, 36, 42, ...) are omitted.

Cf. A208136, A206073, A206284.

t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]];
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]];
TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 250}]];
Map[#[[1]] &, DeleteCases[Table[{z,
   Select[Flatten[Table[CoefficientList[#[[n]], x],
  {n, 1, Length[#]}]] &[Factor[p[z, x]]], # < 0 &]},
  {z, 1, 250}], {_, {}}]]
(* Peter J. C. Moses, Feb 22 2012 *)

A208136 Subsequence of A208135 with numbers that match duplicate factors deleted.

Original entry on oeis.org

9, 33, 35, 39, 49, 57, 65, 129, 133, 135, 147, 159, 161, 183, 201, 215, 225, 235, 237, 249, 259, 267, 287, 291, 303, 371, 385, 393, 413, 417, 423, 427, 459, 489, 497, 519, 525, 527, 537, 543, 573, 579, 591, 605, 609, 615, 633, 651

Offset: 1

Clark Kimberling, Feb 23 2012

nonn

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
  p(1,x) = 1
  p(2,x) = x
  p(3,x) = x + 1
  p(9,x) = x^3 + 1 = (x + 1)*(x^2 - x + 1)
  p(18,x) = x*(x + 1)*(x^2 - x + 1)
  p(33,x) = (x + 1)*(x^4 - x^3 + x^2 - x + 1).
A208135 gives those n for which p(n,x) has a factor containing a negative coefficient; A208136 is a subsequence of A208135 in which, for each p(n,x), there is a factor containing a negative coefficient, and that factor has not already occurred for some p(k,x) with k

			The first few polynomial factors having a negative coefficient are as follows:
  x^2 - x + 1 divides p(n,x) for n=9,18,21,27,36,42,...
  x^4 - x^3 + x^2 - x + 1 divides p(n,x) for n=33,66,...
  x^3 - x^2 + 1 divides p(n,x) for n=35,70,...
  x^4 - x^3 + x^2 + 1 divides p(n,x) for n=39,...
  x^3 - x + 1 divides p(n,x) for n=49,...
  x^4 + x^2 - x + 1 divides p(n,x) for n=57,...
In A208136, the duplicates (such as 18, 21, 27, 36, 42, ...) are omitted.

Cf. A208135, A206073, A206284.

Remove["Global`*"];
t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]];
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]];
TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 900}]];
ans = DeleteCases[Table[{z, Cases[Sign[
       Table[CoefficientList[#[[n]], x], {n, 1, Length[#]}] &[Factor[p[z, x]]]], {_, -1, _}]}, {z, 1, 700}], {_, {}}];
n = 1; While[Length[ans] >= n,
ans = Delete[ans, Map[Take[{#[[1]]}] &, Rest[Position[ans, Flatten[ans[[n]][[2]]]]]]]; n++];
Map[#[[1]] &, ans]
(* Peter J. C. Moses, Feb 22 2012 *)

Previous Showing 21-27 of 27 results.

cp's OEIS Frontend

A208181 Numbers that match polynomials over {0,1} that have a factor containing 3 as a coefficient; see Comments.

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A208182 Numbers that match polynomials over {0,1} that have a factor containing -3 as a coefficient; see Comments.

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A167219 Numbers k such that there exists a positive integer B for which k = Sum_{i=0..m} (B^i)*a_i where the a_i are defined by k = Product_{i=0..m} prime(i+1)^a_i.

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A206285 Numbers that match polynomials not irreducible over the nonnegative integers.

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A206330 Numbers that match polynomials irreducible over the integers.

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A208135 Numbers that match polynomials over {0,1} that have a factor containing a negative coefficient.

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Mathematica

A208136 Subsequence of A208135 with numbers that match duplicate factors deleted.

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