A294893 -id:A294893 - cp's OEIS frontend

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

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).

A294892 Number of proper divisors d of n such that either d=1 or Stern polynomial B(d,x) is reducible.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n=48, its proper divisors are [1, 2, 3, 4, 6, 8, 12, 16, 24]. After 1, the divisors 4, 6, 8, 12, 16 and 24 are not found in A186891, thus a(48) = 1+6 = 7.
For n=50, its proper divisors are [1, 2, 5, 10, 25]. After 1, only 10 is not found in A186891, thus a(50) = 1+1 = 2.

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

Cf. A186891, A283991, A294891, A294893, A294894.

Cf. also A294882, A294902.

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));
A294892(n) = sumdiv(n,d,(dA283991(d)));

a(n) = Sum_{d|n, dA283991(d)).
a(n) + A294891(n) = A032741(n).
a(n) = A294894(n) + A283991(n) - 1.

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).

A294883 Number of divisors of n that are irreducible when their binary expansion is interpreted as polynomial over GF(2).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 09 2017

nonn

Number of terms of A014580 that divide n.

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

Cf. A000005, A014580, A091225, A294881, A294884.
Cf. A091209 (gives a subset of zeros).
Cf. also A234741, A234742, A294893.

A294883(n) = sumdiv(n,d,polisirreducible(Mod(1, 2)*Pol(binary(d))));

a(n) = Sum_{d|n} A091225(d).
a(n) + A294884(n) = A000005(n).
a(n) = A294881(n) + A091225(n).

A294903 Number of divisors of n that are in A257691.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 4, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 4, 3, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 4, 3, 2, 3, 3, 4, 2, 4, 2, 3, 4, 3, 2, 3, 2, 4, 4, 3, 2, 4, 3, 3, 3, 3, 4, 4

Offset: 1

Antti Karttunen, Nov 10 2017

Number of terms of A257691 that divide n.

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

Cf. A000120, A257691, A292257, A294901, A294902, A294904, A294905.

Cf. also A294893.

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

A292257(n) = sumdiv(n,d,(dA294905(n) = (A292257(n) <= hammingweight(n));
A294903(n) = sumdiv(n,d,A294905(d));

a(n) = Sum_{d|n} A294905(d).
a(n) = A294901(n) + A294905(n).
a(n) + A294904(n) = A000005(n).

cp's OEIS Frontend

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

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A294892 Number of proper divisors d of n such that either d=1 or Stern polynomial B(d,x) is reducible.

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

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A294883 Number of divisors of n that are irreducible when their binary expansion is interpreted as polynomial over GF(2).

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A294903 Number of divisors of n that are in A257691.

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