A302786 -id:A302786 - cp's OEIS frontend

A052331 Inverse of A052330; A binary encoding of Fermi-Dirac factorization of n, shown in decimal.

0, 1, 2, 4, 8, 3, 16, 5, 32, 9, 64, 6, 128, 17, 10, 256, 512, 33, 1024, 12, 18, 65, 2048, 7, 4096, 129, 34, 20, 8192, 11, 16384, 257, 66, 513, 24, 36, 32768, 1025, 130, 13, 65536, 19, 131072, 68, 40, 2049, 262144, 258, 524288, 4097, 514, 132, 1048576, 35

Offset: 1

Christian G. Bower, Dec 15 1999

nonn

Every number can be represented uniquely as a product of numbers of the form p^(2^k), sequence A050376. This sequence is a binary representation of this factorization, with a(p^(2^k)) = 2^(i-1), where i is the index (A302778) of p^(2^k) in A050376. Additive with a(p^e) = sum a(p^(2^e_k)) where e = sum(2^e_k) is the binary representation of e and a(p^(2^k)) is as described above. - Franklin T. Adams-Watters, Oct 25 2005 - Index offset corrected by Antti Karttunen, Apr 17 2018

			n = 84 has Fermi-Dirac factorization A050376(5) * A050376(3) * A050376(2) = 7*4*3. Thus a(84) = 2^(5-1) + 2^(3-1) + 2^(2-1) = 16 + 4 + 2 = 22 ("10110" in binary = A182979(84)). - _Antti Karttunen_, Apr 17 2018

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Cf. A050376, A052330, A064547, A302024, A302029, A302776, A302778, A302785, A302786, A302787, A302790, A302784.
Cf. A182979 (same sequence shown in binary).
One less than A064358.
Cf. also A156552.

A052331=a(n)={for(i=1,#n=factor(n)~,n[2,i]>1||next; m=binary(n[2,i]); n=concat(n,Mat(vector(#m-1,j,[n[1,i]^2^(#m-j),m[j]]~)));n[2,i]%=2); n||return(0); m=vecsort(n[1,]); forprime(p=1,m[#m],my(j=0);while(p^2^j>1} \\ M. F. Hasler, Apr 08 2015

up_to_e = 8192;
v050376 = vector(up_to_e);
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_e,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); }; \\ Antti Karttunen, Apr 12 2018

a(1)=0; a(n*A050376(k)) = a(n) + 2^k for a(n) < 2^k, k=0, 1, ... - Thomas Ordowski, Mar 23 2005
From Antti Karttunen, Apr 13 2018: (Start)
a(1) = 0; for n > 1, a(n) = A000079(A302785(n)-1) + a(A302776(n)).
For n > 1, a(n) = A000079(A302786(n)-1) * A302787(n).
a(n) = A064358(n)-1.
A000120(a(n)) = A064547(n).
A069010(a(n)) = A302790(n).
(End)

A223490 Smallest Fermi-Dirac factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 2, 9, 2, 11, 3, 13, 2, 3, 16, 17, 2, 19, 4, 3, 2, 23, 2, 25, 2, 3, 4, 29, 2, 31, 2, 3, 2, 5, 4, 37, 2, 3, 2, 41, 2, 43, 4, 5, 2, 47, 3, 49, 2, 3, 4, 53, 2, 5, 2, 3, 2, 59, 3, 61, 2, 7, 4, 5, 2, 67, 4, 3, 2, 71, 2, 73, 2, 3, 4, 7, 2, 79

Offset: 1

Reinhard Zumkeller, Mar 20 2013

Note that this is not equal to the smallest Fermi-Dirac prime (A050376) dividing n, which is always A020639(n). - Antti Karttunen, Apr 15 2018

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

Cf. A223491, A050376, A028233, A000040 (subsequence).
Cf. A302023, A302024, A302786, A302792.
Cf. also A020639.

Haskell
```
a223490 = head . a213925_row
```

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

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));
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);
A223490(n) = if(1==n,n,A050376(A001511(A052331(n)))); \\ Antti Karttunen, Apr 15 2018

a(n) = A213925(n,1).
A209229(A100995(a(n))) = 1; A010055(a(n)) = 1.
From Antti Karttunen, Apr 15 2018: (Start)
a(1) = 1; and for n > 1, a(n) = A050376(A302786(n)).
a(n) = n / A302792(n).
a(n) = A302023(A020639(A302024(n))).
(End)

A302778 Number of "Fermi-Dirac primes" (A050376) <= n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 16 2018

nonn

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

Partial sums of A302777. A left inverse of A050376.
Cf. A302785, A302786.
Differs from A203967 for the first time at n=64, where a(64) = 23, while A203967(64) = 24.
Cf. also A000720, A025528.

s[n_] := Boole[n > 1 && Length[(f = FactorInteger[n])] == 1 && (e = f[[;; , 2]]) == 2^IntegerExponent[e, 2]]; Accumulate @ Array[s, 100] (* Amiram Eldar, Nov 27 2020 *)

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
s=0; for(n=1,105,s+=A302777(n); print1(s,", "));

from sympy import primepi, integer_nthroot
def A302778(n): return sum(primepi(integer_nthroot(n,1<Chai Wah Wu, Feb 18-19 2025

a(1) = 0; for n > 1, a(n) = A302777(n) + a(n-1).

For all n >= 1, a(A050376(n)) = 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).

A305437 Running index for the lexicographically least irreducible factor when (0,1)-polynomial obtained from the binary expansion of n is factored over Q.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 3, 2, 6, 2, 7, 2, 3, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 3, 2, 13, 2, 14, 2, 3, 2, 5, 2, 15, 2, 3, 2, 16, 2, 17, 2, 3, 2, 18, 2, 5, 2, 3, 2, 19, 2, 20, 2, 3, 2, 21, 2, 22, 2, 3, 2, 4, 2, 23, 2, 24, 2, 25, 2, 26, 2, 3, 2, 27, 2, 28, 2, 29, 2, 30, 2, 4, 2, 31, 2, 32, 2, 33, 2, 10, 2, 4, 2, 34, 2, 3, 2, 35, 2, 36, 2, 3

Offset: 1

Antti Karttunen, Jun 09 2018

nonn

			Numbers 1 .. 6 encode the following (0,1)-polynomials by their binary representation:
  1 -> 1       [Empty factorization]
  2 -> x       [Irreducible, the only and thus also the least factor is x]
  3 -> x + 1   [Irreducible, the least factor is (x+1)]
  4 -> x^2      = (x)(x)   [The least factor (x) occurred already for the first time at n=2, thus a(4) = 2.]
  5 -> x^2 + 1 [Irreducible, the least factor is (x^2 + 1)]
  6 -> x^2 + x  = (x)(x+1) [The least factor (x) occurred already at n=2, thus a(6) = 2.]
Binary representation of 7 is "111", encoding (0,1)-polynomial x^2 + x + 1, which is irreducible over Q, so it is the first time that polynomial occurs as a "smallest" (lexicographically least) irreducible factor, while before it, already four different kinds of "smallest" factors have occurred, thus a(7) = 5.
The second time the same factor occurs as the smallest one is for n=35, whose binary representation "100011" encodes polynomial x^5 + x + 1 = (x^2 + x + 1)(x^3 - x^2 + 1) , thus a(35) = 5 also.
The third time the same factor occurs as the smallest one is for n=49, whose binary representation "110001" encodes polynomial x^5 + x^4 + 1 = (x^2 + x + 1)(x^3 - x + 1), thus a(49) = 5 also.

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

Cf. A206074, A305300.
Cf. A305438 (ordinal transform).
Cf. also A055396, A302786.

allocatemem(2^30);
default(parisizemax,2^31);
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); 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); };
lexleastpolfactor(n) = if(1==n,0,my(fs = factor(Pol(binary(n)))[,1]~); vecsort(fs,pollexcmp)[1]);
v305437 = rgs_transform(vector(up_to,n,lexleastpolfactor(n)));
A305437(n) = v305437[n];

cp's OEIS Frontend

A052331 Inverse of A052330; A binary encoding of Fermi-Dirac factorization of n, shown in decimal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A223490 Smallest Fermi-Dirac factor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A302778 Number of "Fermi-Dirac primes" (A050376) <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A305437 Running index for the lexicographically least irreducible factor when (0,1)-polynomial obtained from the binary expansion of n is factored over Q.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI