A294891 -id:A294891 - cp's OEIS frontend

A294901 Number of proper divisors of n that are in A257691.

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 1, 3, 3, 3, 1, 3, 2, 3, 2, 3, 1, 4, 1, 2, 3, 3, 3, 3, 1, 3, 3, 3, 1, 4, 1, 3, 3, 3, 1, 3, 2, 4, 3, 3, 1, 3, 3, 3, 3, 3, 1, 4, 1, 3, 3, 2, 3, 4, 1, 3, 3, 4, 1, 3, 1, 3, 4, 3, 3, 4, 1, 3, 2, 3, 1, 4, 3, 3, 3, 3, 1, 4, 3, 3, 3, 3, 3, 3, 1, 3, 3, 4, 1, 4, 1, 3, 4, 3, 1, 3, 1, 4, 3, 3, 1, 4, 3, 3, 3, 3, 3, 4

Offset: 1

Antti Karttunen, Nov 10 2017

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

Cf. A000120, A032741, A257691, A292257, A294902, A294903, A294904, A294905.

Cf. also A294891.

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));
A294901(n) = sumdiv(n,d,(dA294905(d));

a(n) = Sum_{d|n, dA294905(d).
a(n) = A294903(n) - A294905(n).
a(n) + A294902(n) = A032741(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.

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).

A294881 Number of proper divisors of n that are irreducible when their binary expansion is interpreted as polynomial over GF(2).

Original entry on oeis.org

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

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, A294882, A294883.

Cf. also A234741, A234742, A294891.

PARI
```
A294881(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, d < n} A091225(d).
a(n) + A294882(n) = A032741(n).
a(n) = A294883(n) - A091225(n).

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A294901 Number of proper divisors of n that are in A257691.

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

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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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A294881 Number of proper divisors of n that are irreducible when their binary expansion is interpreted as polynomial over GF(2).

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