A305437 -id:A305437 - cp's OEIS frontend

A305300 Ordinal transform of A305430, the smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over Q.

1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Jun 09 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206074 (gives the positions of other 1's after the initial one).
Cf. A257000, A305429, A305430, A305437, A305438.
Cf. also A175851.

binPol[n_, x_] := With[{bb = IntegerDigits[n, 2]}, bb.x^Range[Length[bb]-1, 0, -1]];
ip[n_] := If[IrreduciblePolynomialQ[binPol[n, x]], 1, 0];
A305430[n_] := Module[{k = n + 1}, While[ip[k] == 0, k++]; k];
b[_] = 0;
a[n_] := a[n] = With[{t = A305430[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

A257000(n) = polisirreducible(Pol(binary(n)));
A305429(n) = if(n<3,1, my(k=n-1); while(k>1 && !A257000(k),k--); (k));
A305300(n) = if((1==n)||(1==A257000(n)),1,1+(n-A305429(n)));

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A305430(n) = { my(k=1+n); while(!A257000(k),k++); (k); };
v305300 = ordinal_transform(vector(up_to,n,A305430(n)));
A305300(n) = v305300[n];

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], a(n) = 1, otherwise a(n) = 1 + n - A305429(n).

A305438 Number of times the lexicographically least irreducible factor of (0,1)-polynomial (when factored over Q) obtained from the binary expansion of n occurs as the lexicographically least factor for numbers <= n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 4, 14, 1, 15, 1, 16, 5, 17, 2, 18, 1, 19, 6, 20, 1, 21, 1, 22, 7, 23, 1, 24, 3, 25, 8, 26, 1, 27, 1, 28, 9, 29, 1, 30, 1, 31, 10, 32, 2, 33, 1, 34, 1, 35, 1, 36, 1, 37, 11, 38, 1, 39, 1, 40, 1, 41, 1, 42, 3, 43, 1, 44, 1, 45, 1, 46, 2, 47, 4, 48, 1, 49, 12, 50, 1, 51, 1, 52, 13

Offset: 1

Antti Karttunen, Jun 09 2018

nonn

Ordinal transform of A305437.

			Binary representation of 21 is "10101", encoding (0,1)-polynomial x^4 + x^2 + 1 which factorizes over Q as (x^2 - x + 1)(x^2 + x + 1). Factor (x^2 - x + 1) is lexicographically less than factor (x^2 + x + 1) and this is also the first time factor (x^2 - x + 1) occurs as the least one, thus a(21) = 1. Note that although we have the same factor present for n=9, which encodes the polynomial x^3 + 1 = (x + 1)(x^2 - x + 1), it is not the lexicographically least factor in that case.
The next time the same factor occurs as the smallest one is for n=93, which in binary is 1011101, encoding polynomial x^6 + x^4 + x^3 + x^2 + 1 = (x^2 - x + 1)(x^4 + x^3 + x^2 + x + 1). Thus a(93) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206074 (gives a subset of the positions of 1's), A305437.
Cf. A305439.
Cf. also A078898, A302788.

allocatemem(2^30);
default(parisizemax,2^31);
up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
pollexcmp(a,b) = { my(ad = poldegree(a), bd = poldegree(b),e); if(ad != bd, return(sign(ad-bd))); for(i=0,ad,e = polcoeff(a,ad-i) - polcoeff(b,ad-i); if(0!=e, return(sign(e)))); (0); };
Aux305438(n) = if(1==n,0,my(fs = factor(Pol(binary(n)))[,1]~); vecsort(fs,pollexcmp)[1]);
v305438 = ordinal_transform(vector(up_to,n,Aux305438(n)));
A305438(n) = v305438[n];

a(2n) = n.

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A305300 Ordinal transform of A305430, the smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over Q.

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A305438 Number of times the lexicographically least irreducible factor of (0,1)-polynomial (when factored over Q) obtained from the binary expansion of n occurs as the lexicographically least factor for numbers <= n; a(1) = 1.

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