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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.
(history;
published version)
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#12 by Susanna Cuyler at Sat Jun 09 19:00:35 EDT 2018
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#11 by Antti Karttunen at Sat Jun 09 16:58:40 EDT 2018
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#10 by Antti Karttunen at Sat Jun 09 16:52:33 EDT 2018
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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). ThisFactor (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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#9 by Antti Karttunen at Sat Jun 09 16:47:26 EDT 2018
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| NAME
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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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#8 by Antti Karttunen at Sat Jun 09 16:13:34 EDT 2018
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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). This is the first time factor (x^2 - x + 1) occurs, thus a(21) = 1.
The next time the same factor occurs as the smallest one for 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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#7 by Antti Karttunen at Sat Jun 09 15:46:44 EDT 2018
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| LINKS
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Antti Karttunen, <a href="/A305438/b305438.txt">Table of n, a(n) for n = 1..65537</a>
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#6 by Antti Karttunen at Sat Jun 09 15:38:44 EDT 2018
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#5 by Antti Karttunen at Sat Jun 09 15:30:18 EDT 2018
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| COMMENTS
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Ordinal transform of A305437.
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| PROG
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(PARI)
(PARI) allocatemem(2^30);
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| CROSSREFS
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Cf. A206074 (gives a subset of the positions of 1's).), A305437.
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#4 by Antti Karttunen at Sat Jun 09 15:29:02 EDT 2018
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| NAME
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Number of times the "smallest" lexicographically least irreducible factor of (0,1)-polynomial obtained from the binary expansion of n occurs isas the "smallest" such lexicographically least factor for numbers <= n; a(1) = 1.
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| PROG
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(PARI)) allocatemem(2^30);
allocatemem(2^30);
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]);
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#3 by Antti Karttunen at Sat Jun 09 14:23:43 EDT 2018
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| CROSSREFS
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Cf. A206074 (gives a subset of the positions of other 1's after the initial one).
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