Search: a305437 -id:a305437
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A305300
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Ordinal transform of A305430, the smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over Q.
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+10
3
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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
(list;
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listen;
history;
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OFFSET
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1,4
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LINKS
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FORMULA
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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).
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MATHEMATICA
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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];
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PROG
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(PARI)
A257000(n) = polisirreducible(Pol(binary(n)));
A305429(n) = if(n<3, 1, my(k=n-1); while(k>1 && !A257000(k), k--); (k));
(PARI)
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; };
v305300 = ordinal_transform(vector(up_to, n, A305430(n)));
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CROSSREFS
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Cf. A206074 (gives the positions of other 1's after the initial one).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A305438
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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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+10
3
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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a(2n) = n.
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EXAMPLE
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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.
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PROG
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(PARI)
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)));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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