A206284 -id:A206284 - cp's OEIS frontend

A104244 Suppose m = Product_{i=1..k} p_i^e_i, where p_i is the i-th prime number and each e_i is a nonnegative integer. Then we can define P_m(x) = Sum_{i=1..k} e_i*x^(i-1). The sequence is the square array A(n,m) = P_m(n) read by descending antidiagonals.

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 2, 3, 1, 0, 2, 4, 2, 4, 1, 0, 1, 3, 9, 2, 5, 1, 0, 3, 8, 4, 16, 2, 6, 1, 0, 2, 3, 27, 5, 25, 2, 7, 1, 0, 2, 4, 3, 64, 6, 36, 2, 8, 1, 0, 1, 5, 6, 3, 125, 7, 49, 2, 9, 1, 0, 3, 16, 10, 8, 3, 216, 8, 64, 2, 10, 1, 0, 1, 4, 81, 17, 10, 3, 343, 9, 81, 2, 11, 1, 0, 2, 32, 5

Offset: 1

Olaf Voß, Feb 26 2005

From Antti Karttunen, Jul 29 2015: (Start)
The square array A(row,col) is read by downwards antidiagonals as: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
A(n,m) (entry at row=n, column=m) gives the evaluation at x=n of the polynomial (with nonnegative integer coefficients) bijectively encoded in the prime factorization of m. See A206284, A206296 for the details of that encoding. (The roles of variables n and m were accidentally swapped in this description, corrected by Antti Karttunen, Oct 30 2016)
(End)
Each row is a completely additive sequence, row n mapping prime(m) to n^(m-1). - Peter Munn, Apr 22 2022

			a(13) = 3 because 3 = p_1^0 * p_2^1 * p_3^0 * ..., so P_3(x) = 0*x^(1-1) + 1*x^(2-1) + 0*x^(3-1) + ... = x. Hence a(13) = A(3,3) = P_3(3) = 3. [Elaborated by _Peter Munn_, Aug 13 2022]
The top left corner of the array:
0, 1,  1, 2,   1,  2,   1,  3,  2,   2,     1,  3,      1,    2,   2, 4
0, 1,  2, 2,   4,  3,   8,  3,  4,   5,    16,  4,     32,    9,   6, 4
0, 1,  3, 2,   9,  4,  27,  3,  6,  10,    81,  5,    243,   28,  12, 4
0, 1,  4, 2,  16,  5,  64,  3,  8,  17,   256,  6,   1024,   65,  20, 4
0, 1,  5, 2,  25,  6,  125, 3, 10,  26,   625,  7,   3125,  126,  30, 4
0, 1,  6, 2,  36,  7,  216, 3, 12,  37,  1296,  8,   7776,  217,  42, 4
0, 1,  7, 2,  49,  8,  343, 3, 14,  50,  2401,  9,  16807,  344,  56, 4
0, 1,  8, 2,  64,  9,  512, 3, 16,  65,  4096, 10,  32768,  513,  72, 4
0, 1,  9, 2,  81, 10,  729, 3, 18,  82,  6561, 11,  59049,  730,  90, 4
0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4
...

Cf. A000720.
Transpose: A104245.
Main diagonal: A090883.
Row 1: A001222, row 2: A048675, row 3: A090880, row 4: A090881, row 5: A090882, row 10: A054841; and, in the extrapolated table, row 0: A007814, row -1: A195017.
Other completely additive sequences with prime(k) mapped to a function of k include k: A056239, k-1: A318995, k+1: A318994, k^2: A289506, 2^k-1: A293447, k!: A276075, F(k-1): A265753, F(k-2): A265752.
For completely additive sequences with primes p mapped to a function of p, see A001414.
For completely additive sequences where some primes are mapped to 1, the rest to 0 (notably, some ruler functions) see the cross-references in A249344.
For completely additive sequences, s, with primes p mapped to a function of s(p-1) and maybe s(p+1), see A352957.
See the formula section for the relationship to A073133, A206296.
See the comments for the relevance of A206284.
A297845 represents multiplication of the relevant polynomials.
Cf. A090884, A248663, A265398, A265399 for other related sequences.
A167219 lists columns that contain their own column number.

A(n,A206296(k)) = A073133(n,k). [This formula demonstrates how this array can be used with appropriately encoded polynomials. Note that A073133 reads its antidiagonals by ascending order, while here the order is opposite.] - Antti Karttunen, Oct 30 2016
From Peter Munn, Apr 05 2021: (Start)
The sequence is defined by the following identities:
A(n, 3) = n;
A(n, m*k) = A(n, m) + A(n, k);
A(n, A297845(m, k)) = A(n, m) * A(n, k).
(End)

Starting offset changed from 0 to 1 by Antti Karttunen, Jul 29 2015

Name edited (and aligned with rest of sequence) by Peter Munn, Apr 23 2022

A345452 Positive integers with an even number of prime factors (counting repetitions) that sum to an even number.

Original entry on oeis.org

1, 4, 9, 15, 16, 21, 25, 33, 35, 36, 39, 49, 51, 55, 57, 60, 64, 65, 69, 77, 81, 84, 85, 87, 91, 93, 95, 100, 111, 115, 119, 121, 123, 129, 132, 133, 135, 140, 141, 143, 144, 145, 155, 156, 159, 161, 169, 177, 183, 185, 187, 189, 196, 201, 203, 204, 205, 209, 213, 215

Offset: 1

Peter Munn, Jun 20 2021

Numbers with an even number of even prime factors and an even number of odd prime factors.
The representation (as defined in A206284) of polynomials with nonnegative integer coefficients that are in the ideal of the polynomial ring Z[x] generated by x^2+x and 2.
The above property arises because the sequence lists the integers in the multiplicative subgroup of positive rational numbers generated by the squares of primes (A001248) and the products of two consecutive odd primes (A006094\{6}).
The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 4 and 15. For example, A003961(4) = 9, A003961(9) = 25, A003961(15) = 35, 15 * 35 = 525, 525/25 = 21. Alternatively, the sequence may be defined as the closure of A046337 under multiplication by 4.
From the properties of subgroups of the positive rationals we know that if we take an absent positive integer m and divide all terms that are multiples of m by m, we get all the integers in the same subgroup coset as m, and we can expect some of the nice properties here to carry over to the resulting set. Specifically, dividing the even terms by 2 gives all numbers with an odd number of prime factors that sum to an even number; dividing all terms divisible by an odd prime p by p, gives all numbers with an odd number of prime factors that sum to an odd number. The positive integers satisfying the 4th of the 4 possibilities are generated similarly, dividing by 6 (for example).
Numbers whose squarefree part is in A056913.
Term by term, the sequence is one half of its complement within A036349.

			The definition specifies that we count repeated prime factors.
6 = 2 * 3; the sum of these prime factors is 2 + 3 = 5, an odd number; so 6 is not in the sequence.
50 = 2 * 5 * 5 has 3 prime factors and 3 is an odd number; so 50 is not in the sequence.
60 = 2 * 2 * 3 * 5 has 4 prime factors and 4 is an even number; the sum of these factors is 2 + 2 + 3 + 5 = 12, also an even number; so 60 is in the sequence.
1 has 0 prime factors, which sum to 0 (the empty sum). 0 is even, so 1 is in the sequence.

Eric Weisstein's World of Mathematics, Group.
Wikipedia, Polynomial ring.

Cf. A001222, A001414, A003961, A059897, A307150.
Intersection of any 2 of A003159, A028260, A036349.
Other lists that have conditions on the number of odd prime factors: A046337, A072978.
Subsequences: A001248, A006094\{6}, A046315, A056913.

{1}~Join~Select[Range@1000,(s=Flatten[Table@@@FactorInteger[#]];And@@EvenQ@{Length@s,Total@s})&] (* Giorgos Kalogeropoulos, Jun 24 2021 *)

iseven(x) = ((x%2) == 0);
isok(m) = my(f=factor(m)); iseven(sum(k=1, #f~, f[k,1]*f[k,2])) && iseven(sum(k=1, #f~, f[k,2])); \\ Michel Marcus, Jun 24 2021

is(n) = bigomega(n)%2 == 0 && valuation(n, 2)%2 == 0 \\ David A. Corneth, Jun 24 2021

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values())%2 == 0 and sum(p*f[p] for p in f)%2 == 0
print(list(filter(ok, range(1, 216)))) # Michael S. Branicky, Jun 24 2021

{a(n) : n >= 1} = {m >= 1 : A001222(m) mod 2 = A001414(m) mod 2 = 0}.
{A036349(n) : n >= 1} = {a(n) : n >= 1} U {2 * a(n) : n >= 1}.
{A028260(n) : n >= 1} = {a(n) : n >= 1} U {A307150(a(n)) : n >= 1}.
For odd prime p, {A003159(n) : n >= 1} = {a(n) : n >= 1} U {A059897(a(n), p) : n >= 1}.

A206821 Numbers that match irreducible polynomials over {-1,0,1} with leading coefficient 1.

Original entry on oeis.org

2, 3, 7, 8, 10, 14, 16, 18, 21, 23, 29, 31, 35, 41, 42, 44, 48, 50, 54, 56, 60, 62, 66, 70, 72, 76, 78, 80, 82, 84, 86, 88, 93, 97, 99, 103, 109, 111, 115, 117, 123, 125, 129, 131, 137, 141, 143, 147, 153, 155, 159, 161, 165, 167, 171, 173, 179, 183, 186, 188

Offset: 1

Clark Kimberling, Feb 12 2012

nonn

The monic polynomials y(n,x) having coefficients in {-1,0,1} are matched to the set N of positive integers as follows. First, the monic polynomials p(n,x) having coefficients in {0,1} are matched to N as in A206074; i.e., the polynomial x^d(0) + x^d(1) + ... + d(n), where d(i) is 0 or 1 for 0<=i<=n and d(0)=1, matches the binary number d(0)d(1)...d(n). Then monic polynomials having at least one negative coefficient are then inserted among the polynomials p(n,x), as follows: x-1 goes between x and x+1, and for k>1, the polynomials x^k-p(n,x), for 0

n ..... y(n,x) ... irreducible

1 ..... 1 ........ no

2 ..... x ........ yes

3 ..... 1+x ...... yes

4 ..... x^2 ...... no

5 .... -1+x^2 .... no

6 .... -x+x^2 .... no

7 .... -1-x+x^2 .. yes

8 ..... 1+x^2 .... yes

9 ..... x+x^2 .... no

10 .... 1+x+x^2 .. yes

11 .... x^3 ...... no

...

Guide to sequences based on the polynomials y(n,x):

A206822, irreducible

A206829, number of distinct factors

A207187, multiples of x+1

A207188, multiples of x

A207189, multiples of x-1

A207190, multiples of x^2+1

A207191, even: y(n,-x)=y(n,x)

A207192, odd: y(n,-x)=-y(n,x)

Cf. A206073, A206284, A206822.

t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]];
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]];
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]];
q2[n_] := p[n - f[k] + 2];
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
w = {}; Do[n++; If[IrreduciblePolynomialQ[y[[n]]], AppendTo[w, n]], {n, 200}]
w                          (* A206821 *)
Complement[Range[200], w]  (* A206822 *)

A284010 a(n) = least natural number with the same prime signature polynomial p(n,x) has when it is factored over Z. Polynomial p(n,x) has only nonnegative integer coefficients that are encoded in the prime factorization of n.

Original entry on oeis.org

0, 0, 2, 0, 4, 2, 8, 0, 2, 2, 16, 2, 32, 6, 6, 0, 64, 2, 128, 2, 6, 2, 256, 2, 4, 6, 2, 2, 512, 2, 1024, 0, 30, 6, 12, 2, 2048, 6, 6, 2, 4096, 2, 8192, 2, 6, 2, 16384, 2, 8, 2, 30, 2, 32768, 2, 12, 2, 30, 30, 65536, 2, 131072, 6, 6, 0, 60, 2, 262144, 2, 30, 2, 524288, 2, 1048576, 6, 6, 2, 24, 6, 2097152, 2, 2, 6, 4194304, 6, 12, 6, 6, 2, 8388608, 4, 24, 2, 210

Offset: 1

Antti Karttunen, Mar 20 2017

nonn

Let p(n,x) be the completely additive polynomial-valued function such that p(prime(n),x) = x^(n-1) as defined by Clark Kimberling in A206284. To compute a(n), factor p(n,x) over Z and collect the exponents of its irreducible polynomial factors using them as exponents of primes (in Z) as 2^e1 * 3^e2 * 5^e3 * ..., with e1 >= e2 >= e3 >= ...

			For n = 7 = prime(4), the corresponding polynomial is x^3, which factorizes as (x)(x)(x), thus a(7) = 2^3 = 8.
For n = 14 = prime(4) * prime(1), the corresponding polynomial is x^3 + 1, which factorizes as (x + 1)(x^2 - x + 1), thus a(14) = 2^1 * 3^1 = 6.
For n = 90 = prime(3) * prime(2)^2 * prime(1), the corresponding polynomial is x^2 + 2x + 1, which factorizes as (x + 1)^2, thus a(90) = 2^2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A046523, A206284 (positions of 2's), A206442, A277322, A284011, A284012.

Cf. also A260443, A278233, A278243.

\\ After Charles R Greathouse IV's code in A046523 and A277322:
pfps(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2] * 'x^(primepi(f[i, 1])-1)); };
A284010(n) = { if(!bitand(n, (n-1)), 0, my(p=0, f=vecsort(factor(pfps(n))[, 2], ,4)); prod(i=1, #f, (p=nextprime(p+1))^f[i])); }

a(2^n) = 0. [By an explicit convention.]
Other identities. For all n >= 1:
A284011(n) = a(A260443(n)).

A277322 a(n) = number of irreducible polynomial factors (counted with multiplicity) in the polynomial (with nonnegative integral coefficients) constructed from the prime factorization of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 1, 1, 4, 1, 5, 2, 2, 0, 6, 1, 7, 1, 2, 1, 8, 1, 2, 2, 1, 1, 9, 1, 10, 0, 3, 2, 3, 1, 11, 2, 2, 1, 12, 1, 13, 1, 2, 1, 14, 1, 3, 1, 3, 1, 15, 1, 3, 1, 3, 3, 16, 1, 17, 2, 2, 0, 4, 1, 18, 1, 3, 1, 19, 1, 20, 2, 2, 1, 4, 2, 21, 1, 1, 2, 22, 2, 3, 2, 2, 1, 23, 2, 4, 1, 4, 2, 4, 1, 24, 1, 2, 1, 25, 1, 26, 1, 2

Offset: 1

Antti Karttunen, Oct 09 2016

nonn

Let p(n,x) be the completely additive polynomial-valued function such that p(prime(n),x) = x^(n-1) as defined by Clark Kimberling in A206284. Then this sequence is the number of irreducible factors in p(n,x), counted with multiplicity.

			For n = 7 = prime(4), the corresponding polynomial is x^3, which factorizes as (x)(x)(x), thus a(7) = 3.
For n = 14 = prime(4) * prime(1), the corresponding polynomial is x^3 + 1, which factorizes as (x + 1)(x^2 - x + 1), thus a(14) = 2.
For n = 90 = prime(3) * prime(2)^2 * prime(1), the corresponding polynomial is x^2 + 2x + 1, which factorizes as (x + 1)^2, thus a(90) = 2.
pfps(660) = pfps(2^2*3*5*11) = pfps(2^2) + pfps(3) + pfps(5) + pfps(11) = 2 + x + x^2 + x^4 which is irreducible, so a(660) = 1.
For n = 30030 = Product_{i=1..6} prime(i), the corresponding polynomial is x^5 + x^4 + x^3 + x^2 + x + 1, which factorizes as (x+1)(x^2 - x + 1)(x^2 + x + 1), thus a(30030) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..30033

Cf. A206442 (gives the number of irreducible polynomial factors without multiplicity), A206284 (positions of 1's, i.e., irreducible polynomials).

Cf. A000040, A007188, A064989, A260443, A277013.

allocatemem(2^29);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
pfps(n) = if(1==n, 0, if(!(n%2), 1 + pfps(n/2), 'x*pfps(A064989(n))));
A277322 = n -> if(!bitand(n,(n-1)), 0, vecsum(factor(pfps(n))[,2]));
for(n=1, 121121, write("b277322.txt", n, " ", A277322(n)));

pfps(n)=my(f=factor(n)); sum(i=1,#f~, f[i,2] * 'x^(primepi(f[i,1])-1))
A277322(n) = if(1==n, 0, vecsum(factor(pfps(n))[, 2])); \\ Charles R Greathouse IV, test for one added by Antti Karttunen, Oct 09 2016

a(2^n) = 0. [By an explicit convention.]
a(A000040(n)) = n-1.
a(A007188(n)) = n.
a(A260443(n)) = A277013(n).

A277316 Prime-factorization representation of the prime-th Stern-polynomial: a(n) = A260443(A000040(n)).

Original entry on oeis.org

3, 6, 18, 30, 270, 450, 630, 6750, 9450, 22050, 2310, 3543750, 4961250, 53156250, 727650, 173643750, 25467750, 2668050, 40020750, 891371250, 9550406250, 1400726250, 3190703906250, 467969906250, 173423250, 16378946718750, 1715889656250, 245684200781250, 25738344843750, 8497739250, 510510, 6763506750, 66919696593750

Offset: 1

Antti Karttunen, Oct 10 2016

nonn

If the conjecture by Ulas and Ulas is true, then all these terms can be found from A206284 and then this is also a subsequence of A277318.

Antti Karttunen, Table of n, a(n) for n = 1..1028
Maciej Ulas and Oliwia Ulas, On certain arithmetic properties of Stern polynomials, arXiv:1102.5109 [math.CO], 2011.

Cf. A000040, A206284, A260443.
Cf. A277317 (same sequence sorted into ascending order) is a subsequence of A277319.
Differs from A277318 for the first time at n=10, where A277318(10) = 15750, a term which is missing from this sequence.

(define (A277316 n) (A260443 (A000040 n)))

a(n) = A260443(A000040(n)).
Other identities.
For all n >= 1, a(A059305(n)) = A002110(A000043(n)).

A277318 Prime-factorization representation of irreducible (non-constant) Stern-polynomials B(n,x), listed in the order of increasing index n: a(n) = A260443(A186891(n+1)).

Original entry on oeis.org

3, 6, 18, 30, 270, 450, 630, 6750, 9450, 15750, 22050, 2310, 3543750, 4961250, 53156250, 727650, 173643750, 18191250, 25467750, 2668050, 90090, 40020750, 891371250, 9550406250, 212713593750, 1400726250, 3190703906250, 467969906250, 5013963281250, 104053950, 173423250, 16378946718750, 1715889656250, 245684200781250

Offset: 1

Antti Karttunen, Oct 11 2016

nonn

Cf. A186891, A260443.
All terms are included in A206284 and in A277200 (apart from initial 3).
Cf. A277316 (conjectured subsequence), from which this differs for the first time at n=10, where a(10) = 15750 , while A277316(10) = 22050.

A207966 Numbers that match irreducible polynomials over {0,1,2}.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 20, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 49, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 92, 94, 95, 97, 98, 101, 103, 106, 107, 109, 110, 113, 115, 118, 121, 122, 125, 127

Offset: 1

Clark Kimberling, Feb 21 2012

nonn

Each n > 1 matches a polynomial having coefficients in {0,1,2}, determined by the prime factorization of n.
Write n = p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k).
The matching polynomial is then
p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
Identities:
p(m*n) = p(m,x) + p(n,x);
p(m*n) = p(gcd(m,n),x) + p(lcm(m,n),x).
For an analogous enumeration of polynomials over {0,1}, see A206284.
"Irreducible" refers to irreducibility over the field of rational numbers.

			Polynomials having coefficients in {0,1,2} are
matched to the positive integers as follows:
n ... p[n,x] .. irreducible
1 ... 1 ....... no
2 ... 2 ....... no
3 ... x ....... yes
4 ... x+1 ..... yes
5 ... x+2 ..... yes
6 ... 2x ...... yes
7 ... 2x+1 .... yes
8 ... 2x+2 .... yes
9 ... x^2 ..... no
10 .. 1+x^2 ... yes

Cf. A207967, A207968, A206284.

t = Table[IntegerDigits[n, 3], {n, 1, 850}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
  AppendTo[u, n]], {n, 300}]; u  (* A207966 *)
Complement[Range[200], u]        (* A207967 *)
b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
Table[b[n], {n, 1, 50}]          (* A207968 *)

A208179 Numbers that match polynomials with coefficients in {0,1} that have a factor containing 2 as a coefficient; see Comments.

Original entry on oeis.org

141, 177, 183, 237, 282, 354, 366, 427, 474, 555, 564, 573, 663, 669, 699, 708, 711, 717, 723, 732, 741, 753, 813, 849, 854, 871, 885, 909, 923, 933, 948, 951, 1047, 1085, 1110, 1115, 1119, 1128, 1131, 1145, 1146, 1253, 1265, 1299, 1326, 1335

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(141,x) = x^7 + x^3 + + x^2 + 1 = (x + 1)*f(x), where
f(x) = x^6 - x^5 + x^4 - x^3 + 2*x^2 - x + 1. This shows that a factor of p(141,x) has a factor that has 2 as a coefficient. Actually, 141 is the least n for which p(n,x) has a coefficient not in {-1,0,1}.
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 A208179 are disjoint.

			The first five polynomial factors having 2 as a coefficient are indicated here:
  n ..... coefficients of a factor of p(n,x)
  141 ... 1, -1, 2, -1, 1, -1, 1  (see Comments)
  177 ... 1, -1, 1, -1, 2, -1
  183 ... 1, 0, 1, -1, 2, -1, 1
  237 ... 1, -1, 2, -1, 1, 0, 1
  282 ... 1, -1, 2, -1, 1, -1, 1  (same as for n=141)

Cf. A208180, A206073, A206284, A208181, A208182.

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, 1500}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, 2, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 1500}]]]]], {_, {}}]
Map[#[[1]] &, %]
(* Peter J. C. Moses, Feb 22 2012 *)

A208180 Numbers that match polynomials over {0,1} that have a factor containing -2 as a coefficient; see Comments.

Original entry on oeis.org

663, 669, 741, 933, 1326, 1338, 1421, 1482, 1866, 2163, 2181, 2199, 2229, 2247, 2289, 2387, 2469, 2499, 2577, 2589, 2613, 2631, 2643, 2649, 2652, 2661, 2676, 2679, 2757, 2769, 2842, 2949, 2964, 2973, 3115, 3129, 3237, 3241, 3297, 3395

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(663,x) = 1 + x + x^2 + x^4 + x^7 + x^9 = (x + 1)*f(x), where f(x) = 1 + x^2 - x^3 + 2 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8. This show that a factor of p(663,x) has a factor that has -2 as a coefficient. Actually, 663 is the least n for which p(n,x) has a coefficient not in {-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 A208180 are disjoint.

Cf. A208179, A206073, A206284, A208181, A208182.

t = Table[IntegerDigits[n, 2], {n, 1, 4000}];
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, 4000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, -2, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 3600}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208180 *)
(* Peter J. C. Moses, Feb 22 2012 *)

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A277322 a(n) = number of irreducible polynomial factors (counted with multiplicity) in the polynomial (with nonnegative integral coefficients) constructed from the prime factorization of n.

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