A294892 -id:A294892 - cp's OEIS frontend

A294902 Number of proper divisors of n that are in A175526.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 3, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 3, 0, 2, 2, 0, 0, 6, 0, 1, 0, 2, 0, 4, 0, 4, 0, 0, 0, 7, 0, 0, 2, 4, 0, 3, 0, 2, 0, 3, 0, 8, 0, 0, 1, 2, 0, 3, 0, 6, 2, 0, 0, 7, 0, 0, 0, 4, 0, 7, 0, 2, 0, 0, 0, 8, 0, 2, 2, 4, 0, 3, 0, 4, 3, 0, 0, 8, 0, 3, 0, 6, 0, 3, 0, 2, 2, 0, 0, 11

Offset: 1

Antti Karttunen, Nov 10 2017

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A000120, A032741, A175526, A292257, A294901, A294903, A294904, A294905.

Cf. also A294892.

q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] > 2 * DigitCount[n, 2, 1]; a[n_] := DivisorSum[n, 1 &, # < n && q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)

A292257(n) = sumdiv(n,d,(dA294905(n) = (A292257(n) <= hammingweight(n));
A294902(n) = sumdiv(n,d,(dA294905(d)));

a(n) = Sum_{d|n, dA294905(d)).
a(n) = A294904(n) + A294905(n) - 1.
a(n) + A294901(n) = A032741(n).

A294891 Number of proper divisors d of n such that Stern polynomial B(d,x) is irreducible.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 1, 2, 0, 3, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2, 0, 3, 0, 2, 2, 2, 0, 2, 1, 3, 2, 2, 0, 2, 2, 2, 2, 2, 0, 3, 0, 2, 2, 1, 2, 3, 0, 2, 2, 3, 0, 2, 0, 2, 3, 2, 2, 3, 0, 2, 1, 2, 0, 3, 2, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 2, 0, 2, 2, 3, 0, 3, 0, 2, 3

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n=50, with proper divisors [1, 2, 5, 10, 25], 2, 5, and 25 are larger than one and included in A186891, thus a(50) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..22001

Cf. A186891, A283991, A294892, A294893, A294894.
Cf. also A294881, A294901.
Differs from A087624 for the first time at n=50.

ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)));
A283991(n) = polisirreducible(ps(n));
A294891(n) = sumdiv(n,d,(dA283991(d));

a(n) = Sum_{d|n, dA283991(d).
a(n) + A294892(n) = A032741(n).
a(n) = A294893(n) - A283991(n).

A294893 Number of divisors d of n such that Stern polynomial B(d,x) is irreducible.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 2, 3, 2, 2, 2, 1, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 3, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 3, 2, 2, 2, 3, 2, 1, 2, 2, 3, 1, 3, 1, 2, 3

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

Number of terms > 1 of A186891 that divide n.

			For n=25, with divisors [1, 5, 25], both B(5,x) and B(25,x) are irreducible, thus a(25)=2.

Antti Karttunen, Table of n, a(n) for n = 1..22001

Cf. A186891, A283991, A294891, A294892, A294894.
Cf. also A294883.
Differs from A001221 for the first time at n=25.

ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)));
A283991(n) = polisirreducible(ps(n));
A294893(n) = sumdiv(n,d,A283991(d));

a(n) = Sum_{d|n} A283991(d).
a(n) + A294894(n) = A000005(n).
a(n) = A294891(n) + A283991(n).

A294894 Number of divisors d of n such that either d=1 or Stern polynomial B(d,x) is reducible.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 3, 2, 4, 1, 6, 1, 6, 2, 2, 1, 9, 1, 2, 4, 6, 1, 5, 1, 4, 2, 5, 1, 10, 1, 2, 3, 4, 1, 5, 1, 8, 4, 2, 1, 9, 2, 2, 2, 6, 1, 9, 1, 4, 2, 2, 1, 10, 1, 4, 4, 6, 1, 5, 1, 6, 5

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n=25, with divisors [1, 5, 25], both B(5,x) and B(25,x) are irreducible, so only 1 is counted and a(25)=1.

Antti Karttunen, Table of n, a(n) for n = 1..22001

Cf. A186891, A283991, A294891, A294892, A294893.
Cf. also A294884, A294904.
Differs from A033273 for the first time at n=25.

ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)));
A283991(n) = polisirreducible(ps(n));
A294894(n) = sumdiv(n,d,(0==A283991(d)));

a(n) = Sum_{d|n} (1-A283991(d)).
a(n) + A294893(n) = A000005(n).
a(n) = 1 + A294892(n) - A283991(n).

A294882 Number of proper divisors of n that are not irreducible when their binary expansion is interpreted as polynomial over GF(2).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 3, 1, 3, 1, 4, 1, 1, 1, 5, 2, 1, 2, 3, 1, 5, 1, 4, 1, 2, 2, 6, 1, 1, 1, 6, 1, 4, 1, 3, 4, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 1, 2, 1, 9, 1, 1, 3, 5, 2, 4, 1, 4, 2, 5, 1, 9, 1, 1, 3, 3, 1, 4, 1, 8, 3, 1, 1, 8, 3, 2, 2, 5, 1, 9, 1, 4, 1, 1, 2, 9, 1, 3, 3, 6, 1, 5, 1, 5, 5

Offset: 1

Antti Karttunen, Nov 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A032741, A091225, A294881, A294884.

Cf. also A234741, A234742, A294892.

PARI
```
A294882(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA091225(d)).
a(n) + A294881(n) = A032741(n).
For n > 1, a(n) = A294884(n) + A091225(n) - 1.

cp's OEIS Frontend

A294902 Number of proper divisors of n that are in A175526.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294891 Number of proper divisors d of n such that Stern polynomial B(d,x) is irreducible.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A294893 Number of divisors d of n such that Stern polynomial B(d,x) is irreducible.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A294894 Number of divisors d of n such that either d=1 or Stern polynomial B(d,x) is reducible.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A294882 Number of proper divisors of n that are not irreducible when their binary expansion is interpreted as polynomial over GF(2).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula