A052331 -id:A052331 - cp's OEIS frontend

A302023 Permutation of natural numbers mapping ordinary factorization to "Fermi-Dirac factorization": a(1) = 1, a(2n) = 2*A300841(a(n)), a(A003961(n)) = A300841(a(n)).

1, 2, 3, 6, 4, 8, 5, 24, 12, 10, 7, 30, 9, 14, 15, 120, 11, 40, 13, 42, 21, 18, 16, 168, 20, 22, 60, 54, 17, 56, 19, 840, 27, 26, 28, 210, 23, 32, 33, 216, 25, 72, 29, 66, 84, 34, 31, 1080, 35, 70, 39, 78, 37, 280, 36, 264, 48, 38, 41, 270, 43, 46, 108, 7560, 44, 88, 47, 96, 51, 90, 49, 1512, 53, 50, 105, 102, 45, 104, 59, 1320

Offset: 1

Antti Karttunen, Apr 15 2018

nonn

See comments and additional formulas in A302024.

Cf. A302024 (inverse).
Cf. A050376, A052330, A064989, A156552, A300841.
Cf. also A091202, A302025.

up_to = 32768;
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));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
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); };
A300841(n) = A052330(2*A052331(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A302023(n) = if(1==n,n,if(!(n%2),2*A300841(A302023(n/2)),A300841(A302023(A064989(n)))));

a(1) = 1; a(2n) = 2*A300841(a(n)), a(2n+1) = A300841(a(A064989(2n+1))). [corrected Jun 10 2018]
a(n) = A052330(A156552(n)).
a(A000040(n)) = A050376(n).

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).

A182979 Fermi-Dirac representation of n. Let n have factorization p1^(2^e1) * p2^(2^e2) * ... * pr^(2^er), where each factor is in A050376. The number n is represented by a binary string that indicates which terms of A050376 appear in the factorization of n.

Original entry on oeis.org

0, 1, 10, 100, 1000, 11, 10000, 101, 100000, 1001, 1000000, 110, 10000000, 10001, 1010, 100000000, 1000000000, 100001, 10000000000, 1100, 10010, 1000001, 100000000000, 111, 1000000000000, 10000001, 100010, 10100, 10000000000000, 1011, 100000000000000, 100000001, 1000010, 1000000001, 11000, 100100

Offset: 1

Daniel Forgues, Feb 10 2011, Feb 13 2011

nonn

Every number has a unique representation as a product of terms from A050376. - N. J. A. Sloane, Feb 11 2011
The "Fermi-Dirac factorization" of n, i.e., the factorization of n into prime powers of the form p_k^(2^e_k), e_k >= 0, (A050376) allows each of those prime powers to be used at most once, since this corresponds to the binary representation of the exponents of the prime powers p^a of the "Bose-Einstein factorization" of n, i.e., the classic prime factorization of n. (Cf. A050376 comments.)
The prime powers of the form p_k^(2^e_k), e_k >= 0 (A050376) might be called "Fermi-Dirac primes" since they may appear at most once (thus raised to powers 0 or 1) in the "Fermi-Dirac factorization" of n. Compare with the classic prime factorization of n, which might be called the "Bose-Einstein factorization" of n, where the primes (which might be called "Bose-Einstein primes") may appear any number of times >= 0.
In the "Fermi-Dirac representation" of n, if a given prime power with powers of two as exponents does not appear in the factorization of n into prime powers with powers of two as exponents, we use 0 as a placeholder; otherwise, we use 1 to indicate that the given prime power with powers of two as exponents does appear in the "Fermi-Dirac factorization" of n.
In the base-b representation of n, we do not show the leading 0's, except for 0 where it is more convenient to show it than to show nothing. Similarly, for the "Fermi-Dirac representation" of n, we do not show the leading 0's, except for 0, which is the representation of 1, where it is more convenient to show it than to show nothing.
The limit of the supremum of the number of "binary digits" of the representation of n is asymptotic to the number of primes up to n, i.e., n/log(n), making this representation absolutely impractical!
See A052330 for the numbers having representation as 0, 1, 10, 11, 100, 101, 110, 111, ... which is an ordering of the positive integers. (Cf. OEIS Wiki page.)
Let n have factorization (f_r)^g_r * ... * (f_2)^g_2 * (f_1)^g_1, where f_i is the i-th prime power of the form p_k^(2^e_k), e_k >= 0 (A050376, A302778); then a(n) = Sum_{i=1..r} g_i * 2^(i-1).
The number of 1's in a(n) is the number of terms of A050376 dividing n with odd maximal exponent. For example, if n=96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus only 2, 3 and 16 divide n with odd maximal exponents. Therefore, the number of 1's in a(96) is 3. Moreover, since 2=A050376(1), 3=A050376(2) and 16=A050376(9), then 1's appear in positions 1,2,9 from the right. - Vladimir Shevelev, Nov 02 2013

			"Fermi-Dirac factorizations" (cf. A050376 examples, here with exponents of "Fermi-Dirac primes" being either 0 or 1):
6 = 3*2 = 3^1*2^1, so a(6) = 11;
8 = 4*2 = 4^1*3^0*2^1, so a(8) = 101;
20 = 5*4 = 5^1*4^1*3^0*2^0, so a(20) = 1100;
24 = 4*3*2 = 4^1*3^1*2^1, so a(24) = 111;
27 = 9*3 = 9^1*7^0*5^0*4^0*3^1*2^0, so a(27) = 100010;
32 = 16*2 = 16^1*13^0*11^0*9^0*7^0*5^0*4^0*3^0*2^1, so a(32) = 100000001;
64 = 16*4 = 16^1*13^0*11^0*9^0*7^0*5^0*4^1*3^0*2^0, so a(64) = 100000100;
108 = 9*4*3 = 9^1*7^0*5^0*4^1*3^1*2^0, so a(108) = 100110;
120 = 5*4*3*2 = 5^1*4^1*3^1*2^1, so a(120) = 1111;
...

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, "Fermi-Dirac representation" of n
OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"

Cf. A050376, A052330, A052331, A302778.

nn=24; p=Select[Range[nn], PrimeQ]; Do[p=Select[Union[p,p^2], #<=nn&], {Floor[Log[2,Log[2,nn]]]}]; Table[m=n; FromDigits[Table[If[Mod[m,i]==0, m=m/i; 1, 0], {i,Reverse[p]}]],{n,nn}]

Let q_1,q_2,q_3,... be consecutive terms of A050376 and n = q_1^a_1 * q_2^a_2 *...* q_r^a_r, where a_i = 0 or 1. Then a(n) = a_1 + 10*a_2 + ... +10^(r-1)*a_r. For example, since 30 = 2^1 * 3^1 * 4^0 * 5^1, then a(30)= 1 + 10 + 1000 = 1011. - Vladimir Shevelev, Nov 02 2013

a(n) = A007088(A052331(n)). - Antti Karttunen, Apr 17 2018

Clearer definition from T. D. Noe, Feb 11 2011

Edited by N. J. A. Sloane, Jul 21 2018

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];

A302791 A filter sequence for Fermi-Dirac factorization: restricted growth sequence transform of A046523(A302024(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 15 2018

nonn

For all i, j: a(i) = a(j) => A064547(i) = A064547(j).
For all i, j: a(i) = a(j) => A302790(i) = A302790(j).
See also comments in A302024.

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

Cf. A037445, A046523, A050376 (gives the positions of 2's), A052331, A064547, A293442, A302024, A302787, A302790.

allocatemem(2^30);
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); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A302024(n) = A005940(1+A052331(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux302791(n) = A046523(A302024(n));
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302791(n))),"b302791.txt");

A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1, ..., y_k) is f(y_1) * ... * f(y_k).

			45 is the FDH number of (6,4), which has GCD 2, so a(45) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

A left inverse of A050376.

Cf. A052331, A056239, A064547, A213925, A289508, A289509, A290103, A299755, A299757, A319825.

nn=200;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
GCD@@@Table[Reverse[FDfactor[n]/.FDrules],{n,nn}]

A319826(n) = { my(i=1,g=0,x=A052331(n)); while(x,if(x%2,g = gcd(g,i)); x>>=1; i++); (g); }; \\ (Uses the program given in A052331) - Antti Karttunen, Feb 18 2023

For all n >= 1, a(A050376(n)) = n. - Antti Karttunen, Feb 18 2023

Secondary definition added by Antti Karttunen, Feb 18 2023

A302776 a(1) = 1; for n>1, a(n) = n/(largest Fermi-Dirac factor of n).

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, 6, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 6, 5, 8, 3, 2, 1, 12, 1, 2, 7, 4, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 12, 5, 2, 3, 8, 1, 10, 7, 4, 3, 2, 5, 6, 1, 2, 9, 4, 1, 6, 1, 8, 15

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

For n > 1, a(n) = the smallest positive number d such that n/d is a "Fermi-Dirac prime", a term of A050376.

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

Cf. A050376, A052331, A223491, A302023, A302024, A302777, A302792.
Cf. A084400 (gives the positions of 1's).
Cf. also A052126, A284600.

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

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A302776(n) = if(1==n,n,fordiv(n, d, if(A302777(n/d), return(d))));

a(n) = n / A223491(n).

a(n) = A302023(A052126(A302024(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];

A302792 a(1) = 1; for n>1, a(n) = n/(smallest Fermi-Dirac factor of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 4, 1, 5, 1, 4, 1, 7, 5, 1, 1, 9, 1, 5, 7, 11, 1, 12, 1, 13, 9, 7, 1, 15, 1, 16, 11, 17, 7, 9, 1, 19, 13, 20, 1, 21, 1, 11, 9, 23, 1, 16, 1, 25, 17, 13, 1, 27, 11, 28, 19, 29, 1, 20, 1, 31, 9, 16, 13, 33, 1, 17, 23, 35, 1, 36, 1, 37, 25, 19, 11, 39, 1, 16, 1, 41, 1, 28, 17, 43, 29, 44, 1, 45, 13, 23, 31, 47, 19, 48, 1

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

The positive integers that are absent from this sequence are A036554, integers that have 2 as a Fermi-Dirac factor. - Peter Munn, Apr 23 2018

a(n) is the largest aliquot infinitary divisor of n, for n > 1 (cf. A077609). - Amiram Eldar, Nov 19 2022

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

Cf. A001511, A036554, A050376, A052331, A077609, A223490, A302023, A302024, A302776, A300841.
Cf. A084400 (gives the positions of 1's).
Cf. also A032742.

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := n / Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 27 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))));
A302792(n) = (n/A223490(n));

a(n) = {if(n==1, 1, my(f = factor(n)); for(i=1, #f~, f[i,1] = f[i,1]^(1<Amiram Eldar, Nov 19 2022

a(n) = n / A223490(n).

a(n) = A300841(A302023(A032742(A302024(n)))).

cp's OEIS Frontend

A302023 Permutation of natural numbers mapping ordinary factorization to "Fermi-Dirac factorization": a(1) = 1, a(2n) = 2*A300841(a(n)), a(A003961(n)) = A300841(a(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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A182979 Fermi-Dirac representation of n. Let n have factorization p1^(2^e1) * p2^(2^e2) * ... * pr^(2^er), where each factor is in A050376. The number n is represented by a binary string that indicates which terms of A050376 appear in the factorization of 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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A302791 A filter sequence for Fermi-Dirac factorization: restricted growth sequence transform of A046523(A302024(n)).

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A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

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A302776 a(1) = 1; for n>1, a(n) = n/(largest Fermi-Dirac factor of 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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