A302789 -id:A302789 - cp's OEIS frontend

A223491 Largest Fermi-Dirac factor of n.

1, 2, 3, 4, 5, 3, 7, 4, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 4, 25, 13, 9, 7, 29, 5, 31, 16, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 9, 16, 13, 11, 67, 17, 23, 7, 71, 9

Offset: 1

Reinhard Zumkeller, Mar 20 2013

nonn

Greatest Fermi-Dirac factor of n: Largest divisor of n of the form p^(2^k), for some prime p and k >= 0, with a(1) = 1. Thus for n > 1, the largest term of A050376 that divides n. - Antti Karttunen, Apr 13 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
OEIS Wiki, "Fermi-Dirac representation" of n

Cf. A223490, A050376, A034699, A000040 (subsequence), A302776, A302785, A302789 (ordinal transform).

Cf. also A006530, A034699.

Haskell
```
a223491 = last . a213925_row
```

f[p_, e_] := p^(2^Floor[Log2[e]]); a[n_] := Max @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)

ispow2(n) = (n && !bitand(n,n-1));
A223491(n) = if(1==n,n,fordiv(n, d, if(ispow2(isprimepower(n/d)), return(n/d)))); \\ Antti Karttunen, Apr 13 2018

a(n) = A213925(n,A064547(n)).
A209229(A100995(a(n))) = 1; A010055(a(n)) = 1.
From Antti Karttunen, Apr 13 2018: (Start)
a(1) = 1; for n > 1, a(n) = A050376(A302785(n)).
a(n) = n/A302776(n).
(End)

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

Original entry on oeis.org

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).

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.

A302788 Number of times the smallest Fermi-Dirac factor of n is the smallest Fermi-Dirac factor for numbers <= n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 3, 1, 1, 6, 1, 2, 4, 7, 1, 8, 1, 9, 5, 3, 1, 10, 1, 11, 6, 12, 2, 4, 1, 13, 7, 14, 1, 15, 1, 5, 3, 16, 1, 8, 1, 17, 9, 6, 1, 18, 4, 19, 10, 20, 1, 11, 1, 21, 2, 7, 5, 22, 1, 8, 12, 23, 1, 24, 1, 25, 13, 9, 3, 26, 1, 6, 1, 27, 1, 14, 7, 28, 15, 29, 1, 30, 4, 10, 16, 31, 8, 32, 1, 33, 2, 11, 1, 34, 1, 35, 17

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

Ordinal transform of A223490, or equally, of A302786.

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

Cf. A001511, A050376, A052331, A223490, A302786, A302789.
Cf. A084400 (gives the positions of 1's).
Cf. also A078898.

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; };
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));
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)));
v302788 = ordinal_transform(vector(up_to,n,A302786(n)));
A302788(n) = v302788[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).

A303759 Number of times the largest prime power factor of n (A034699) is largest prime power factor for numbers <= n; a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of A034699.

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

Cf. A000961 (positions of ones), A034699.

Cf. also A078899, A284600, A302789.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= max(1, seq(i[1]^i[2], i=ifactors(n)[2]));
      b(t):= b(t)+1
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 30 2018

f[n_] := Max[Power @@@ FactorInteger[n]];
b[_] = 0;
a[n_] := With[{t = f[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Jan 03 2022 *)

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; };
A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d))));
v303759 = ordinal_transform(vector(up_to,n,A034699(n)));
A303759(n) = v303759[n];

cp's OEIS Frontend

A223491 Largest Fermi-Dirac factor of n.

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

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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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A302788 Number of times the smallest Fermi-Dirac factor of n is the smallest Fermi-Dirac factor for numbers <= n; a(1) = 1.

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

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A303759 Number of times the largest prime power factor of n (A034699) is largest prime power factor for numbers <= n; a(1) = 1.

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