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.

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

Author

Antti Karttunen, Jun 09 2018

Keywords

• nonn

Comments

Ordinal transform of A305437.

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A305439.

Cf. also A078898, A302788.

Programs

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)));
A305438(n) = v305438[n];

Formula

a(2n) = n.