A302788 -id:A302788 - cp's OEIS frontend

A302786 Index of the smallest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223490(n)).

0, 1, 2, 3, 4, 1, 5, 1, 6, 1, 7, 2, 8, 1, 2, 9, 10, 1, 11, 3, 2, 1, 12, 1, 13, 1, 2, 3, 14, 1, 15, 1, 2, 1, 4, 3, 16, 1, 2, 1, 17, 1, 18, 3, 4, 1, 19, 2, 20, 1, 2, 3, 21, 1, 4, 1, 2, 1, 22, 2, 23, 1, 5, 3, 4, 1, 24, 3, 2, 1, 25, 1, 26, 1, 2, 3, 5, 1, 27, 4, 28, 1, 29, 2, 4, 1, 2, 1, 30, 1, 5, 3, 2, 1, 4, 1, 31, 1, 6, 3, 32, 1, 33, 1, 2

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

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

A left inverse of A050376.
Cf. A001511, A052331, A223490, A302778, A302785, A302787, A302788 (ordinal transform), A302789.
Cf. also A055396.

nn = 105; t = {}; k = 1;
While[lim = nn^(1/k); lim > 2,
     t = Join[t, Prime[Range[PrimePi[lim]]]^k]; k = 2 k];
A050376 = Union[t];
A223490[n_] := Table[{p, e} = pe; p^(2^IntegerExponent[e, 2]), {pe, FactorInteger[n]}] // Min;
a[n_] := If[n == 1, 0, FirstPosition[A050376, A223490[n]][[1]]];
Array[a, nn] (* Jean-François Alcover, Jan 08 2022, after T. D. Noe in A050376 *)

up_to = 65537;
v050376 = vector(up_to);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A001511(n) = 1+valuation(n,2);
A302786(n) = if(1==n,0,A001511(A052331(n)));

a(n) = A302778(A223490(n)).
a(1) = 0; for n > 1, a(n) = A001511(A052331(n)).
For n >= 1, a(A050376(n)) = n.
For n > 1, A050376(a(n)) = A223490(n).

A302789 Number of times the largest Fermi-Dirac factor of n is the largest Fermi-Dirac factor for numbers <= n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 2, 3, 2, 5, 4, 1, 2, 3, 6, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 6, 5, 7, 3, 2, 1, 7, 1, 2, 7, 4, 5, 6, 1, 4, 3, 8, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 9, 5, 2, 3, 8, 1, 9, 7, 4, 3, 2, 5, 6, 1, 2, 9, 4, 1, 6, 1, 8, 10

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

Ordinal transform of A223491, or equally, of A302785.

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

Cf. A050376, A052331, A223491, A302785, A302788.
Cf. A084400 (gives the positions of 1's).
Cf. also A078899.

f[n_] := Max@Table[{p, e} = pe; p^(2^(Length[IntegerDigits[e, 2]]-1)), {pe, FactorInteger[n]}];
b[_] = 1;
a[n_] := a[n] = With[{t = f[n]}, b[t]++];
Array[a, 105] (* Jean-François Alcover, Dec 18 2021 *)

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; };
ispow2(n) = (n && !bitand(n, n-1));
A223491(n) = if(1==n,n,fordiv(n, d, if(ispow2(isprimepower(n/d)), return(n/d))));
v302789 = ordinal_transform(vector(up_to,n,A223491(n)));
A302789(n) = v302789[n];

A302785 Index of the largest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223491(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 3, 8, 5, 4, 9, 10, 6, 11, 4, 5, 7, 12, 3, 13, 8, 6, 5, 14, 4, 15, 9, 7, 10, 5, 6, 16, 11, 8, 4, 17, 5, 18, 7, 6, 12, 19, 9, 20, 13, 10, 8, 21, 6, 7, 5, 11, 14, 22, 4, 23, 15, 6, 9, 8, 7, 24, 10, 12, 5, 25, 6, 26, 16, 13, 11, 7, 8, 27, 9, 28, 17, 29, 5, 10, 18, 14, 7, 30, 6, 8, 12, 15, 19, 11, 9, 31, 20, 7, 13, 32, 10, 33, 8, 5

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

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

A left inverse of A050376.
Cf. A052331, A223491, A240535, A302778, A302786, A302788, A302789 (ordinal transform).
Cf. also A061395.

up_to = 65537;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A302785(n) = if(1==n,0, my(e); fordiv(n, d, if(ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(e, return(e), print("v050376 too short!"); return(1/0)))));

a(n) = A302778(A223491(n)).
For n > 1, A050376(a(n)) = A223491(n).
For n >= 1, a(A050376(n)) = n.

A302787 a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 5, 1, 9, 1, 3, 1, 17, 5, 1, 1, 33, 1, 3, 9, 65, 1, 7, 1, 129, 17, 5, 1, 11, 1, 257, 33, 513, 3, 9, 1, 1025, 65, 13, 1, 19, 1, 17, 5, 2049, 1, 129, 1, 4097, 257, 33, 1, 35, 9, 21, 513, 8193, 1, 7, 1, 16385, 3, 65, 17, 67, 1, 129, 1025, 25, 1, 37, 1, 32769, 2049, 257, 5, 131, 1, 33, 1, 65537, 1, 11, 65, 131073, 4097, 69, 1, 41, 9

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

After n=1, differs from A240535 (which gives the same terms, but with mirrored binary expansion) for the first time at n=30, where a(30) = 11, while A240535(30) = 13. Note how 11 = "1011" and 13 = "1101" in binary.

For all i, j: a(i) = a(j) => A302791(i) = A302791(j).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000265, A050376, A052331, A240535, A302786, A302788, A302789, A302790, A302791.

Cf. also A246277.

up_to = 8192;
v050376 = vector(up_to);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A000265(n) = (n/2^valuation(n, 2));
A302787(n) = if(1==n,0,A000265(A052331(n)));

a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).
For n > 1, a(n) = A030101(A240535(n)).
For n >= 1, A069010(a(n)) = A302790(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.

cp's OEIS Frontend

A302786 Index of the smallest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223490(n)).

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A302789 Number of times the largest Fermi-Dirac factor of n is the largest Fermi-Dirac factor for numbers <= n; a(1) = 1.

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A302785 Index of the largest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223491(n)).

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A302787 a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).

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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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