A302024 -id:A302024 - cp's OEIS frontend

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

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");

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119

Offset: 1

Marc LeBrun

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to Heinz numbers.

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))

a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016

from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

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

Original entry on oeis.org

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)

A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 32, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 243, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47, 324

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

E.g. we have the following identities: A000040(n) = a(A014580(n)), A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)), A091227(n) = A049084(a(n)), A091247(n) = A066247(a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091202.
Several variants exist: A235042, A091205, A106443, A106445, A106447.
Cf. also A000005, A000040, A001221, A001222, A004247, A008683, A010051, A014580, A048720, A049084, A066247, A091219, A091220, A091221, A091222, A091225, A091227, A091247, A234741, A234742, A245703, A245704, A278233.
Cf. also A302024, A302026, A305418, A305428 for other similar permutations.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A091203(n) = if(n<=1,n,if(!(n%2),2*A091203(n/2),A003961(A091203(A305422(n))))); \\ Antti Karttunen, Jun 10 2018

a(0)=0, a(1)=1. For n's coding an irreducible polynomial ir_i, that is if n=A014580(i), we have a(n) = A000040(i) and for composite polynomials a(ir_i X ir_j X ...) = p_i * p_j * ..., where p_i = A000040(i) and X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).
Other identities. For all n >= 1, the following holds:
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials, A014580, to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers. The permutations A091205, A106443, A106445, A106447, A235042 and A245704 have the same property.]
From Antti Karttunen, Jun 10 2018: (Start)
For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A003961(a(A305422(n))).
a(n) = A005940(1+A305418(n)) = A163511(A305428(n)).
A046523(a(n)) = A278233(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)

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)).

Original entry on oeis.org

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).

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))).

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)))).

A302790 Number of runs of consecutive Fermi-Dirac factors of n (the runs are separated by gaps between indices of factors): a(n) = A069010(A052331(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

			n = 84 has Fermi-Dirac factorization as A050376(2) * A050376(3) * A050376(5) = 3*4*7. Because there is a gap between A050376(3) and A050376(5), the factors occur in two separate runs (3*4 and 7), thus a(84) = 2.

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

Cf. A052331, A069010, A302787.

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); };
A069010(n) = ((1 + (hammingweight(bitxor(n, n>>1)))) >> 1); \\ From A069010
A302790(n) = A069010(A052331(n));

a(n) = A069010(A052331(n)).
a(n) = A069010(A302787(n)).
a(n) = A001221(A302024(n)).
For all n >= 1, a(n) <= A064547(n).

cp's OEIS Frontend

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

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A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

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

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A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers.

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A223490 Smallest Fermi-Dirac factor of n.

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

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