A294884 -id:A294884 - cp's OEIS frontend

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

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.

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).

A305815 Restricted growth sequence transform of A305814, a filter sequence constructed from the GF(2)[X]-factorization signatures of the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 6, 7, 8, 2, 9, 2, 5, 6, 10, 11, 12, 2, 13, 14, 5, 15, 16, 3, 5, 17, 9, 15, 16, 2, 18, 5, 19, 20, 21, 2, 5, 22, 23, 2, 24, 15, 9, 25, 26, 2, 27, 7, 9, 28, 9, 15, 29, 14, 16, 22, 26, 2, 30, 2, 5, 9, 31, 32, 33, 2, 34, 20, 35, 15, 36, 2, 5, 37, 9, 5, 38, 15, 39, 40, 5, 41, 42, 43, 26, 5, 16, 15, 44, 45, 46, 5, 5, 8, 47, 2, 12, 48, 49, 41, 50

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A278233, A305788, A305813, A305814.

Cf. also A305795.

\\ Needs also code from A305788:
up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
A305814(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A305788(d)-1))); (m); };
v305815 = rgs_transform(vector(up_to, n, A305814(n)));
A305815(n) = v305815[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A294883(i) = A294883(j).
  a(i) = a(j) => A294884(i) = A294884(j).

cp's OEIS Frontend

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

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

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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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A305815 Restricted growth sequence transform of A305814, a filter sequence constructed from the GF(2)[X]-factorization signatures of the divisors of n.

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PARI

Formula