A300841 -id:A300841 - cp's OEIS frontend

A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2A300841(n)) = 2a(n), a(A300841(n)) = A003961(a(n)).

1, 2, 3, 5, 7, 4, 11, 6, 13, 10, 17, 9, 19, 14, 15, 23, 29, 22, 31, 25, 21, 26, 37, 8, 41, 34, 33, 35, 43, 12, 47, 38, 39, 46, 49, 55, 53, 58, 51, 18, 59, 20, 61, 65, 77, 62, 67, 57, 71, 74, 69, 85, 73, 28, 91, 30, 87, 82, 79, 27, 83, 86, 121, 95, 119, 44, 89, 115, 93, 50, 97, 42, 101, 94, 111, 145, 143, 52, 103, 133, 107, 106, 109, 45, 161

Offset: 1

Antti Karttunen, Apr 15 2018

nonn

Because "Fermi-Dirac factorization" is fundamentally different from ordinary prime factorization (as no exponents larger than 1 are allowed) this pair of permutations mapping between them is not always very intuitive. For example, we have ("as expected") A302776(n) = A302023(A052126(A302024(n))), while on the other hand, we have A302792(n) = A300841(A302023(A032742(A302024(n)))), where an additional shift-operator A300841 is needed for "correction".

Cf. A302023 (inverse).

Cf. A050376, A052331, A003961, A005940, A300840, A300841, A302791.

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));
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));

a(n) = A005940(1+A052331(n)).
a(A050376(n)) = A000040(n).
A001221(a(n)) = A302790(n).
A001222(a(n)) = A064547(n).

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).

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

A036554 Numbers whose binary representation ends in an odd number of zeros.

Original entry on oeis.org

2, 6, 8, 10, 14, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 50, 54, 56, 58, 62, 66, 70, 72, 74, 78, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 114, 118, 120, 122, 126, 128, 130, 134, 136, 138, 142, 146, 150, 152, 154, 158, 160, 162, 166, 168, 170, 174

Offset: 1

Tom Verhoeff

Fraenkel (2010) called these the "dopey" numbers.
Also n such that A035263(n)=0 or A050292(n) = A050292(n-1).
Indices of even numbers in A033485. - Philippe Deléham, Mar 16 2004
a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even. - Philippe Deléham, Mar 16 2004
Indices of even numbers in A007913, in A001511. - Philippe Deléham, Mar 27 2004
This sequence consists of the increasing values of n such that A097357(n) is even. - Creighton Dement, Aug 14 2004
Numbers with an odd number of 2's in their prime factorization (e.g., 8 = 2*2*2). - Mark Dow, Sep 04 2007
Equals the set of natural numbers not in A003159 or A141290. - Gary W. Adamson, Jun 22 2008
Represents the set of CCW n-th moves in the standard Tower of Hanoi game; and terms in even rows of a [1, 3, 5, 7, 9, ...] * [1, 2, 4, 8, 16, ...] multiplication table. Refer to the example. - Gary W. Adamson, Mar 20 2010
Refer to the comments in A003159 relating to A000041 and A174065. - Gary W. Adamson, Mar 21 2010
If the upper s-Wythoff sequence of s is s, then s=A036554. (See A184117 for the definition of lower and upper s-Wythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A036554 is the limit when the upper s-Wythoff operation is iterated.  For example, starting with s=(1,4,9,16,...) = (n^2), we obtain lower and upper s-Wythoff sequences
  a=(1,3,4,5,6,8,9,10,11,12,14,...) = A184427;
  b=(2,7,12,21,31,44,58,74,...) = A184428.
  Then putting s=a and repeating the operation gives
b'=(2,6,8,10,13,17,20,...), which has the same first four terms as A036554. - Clark Kimberling, Jan 14 2011
Or numbers having infinitary divisor 2, or the same, having factor 2 in Fermi-Dirac representation as a product of distinct terms of A050376. - Vladimir Shevelev, Mar 18 2013
Thus, numbers not in A300841 or in A302792. Equally, sequence 2*A300841(n) sorted into ascending order. - Antti Karttunen, Apr 23 2018

			From _Gary W. Adamson_, Mar 20 2010: (Start)
Equals terms in even numbered rows in the following multiplication table:
(rows are labeled 1,2,3,... as with the Towers of Hanoi disks)
   1,  3,  5,  7,  9, 11, ...
   2,  6, 10, 14, 18, 22, ...
   4, 12, 20, 28, 36, 44, ...
   8, 24, 40, 56, 72, 88, ...
   ...
As shown, 2, 6, 8, 10, 14, ...; are in even numbered rows, given the row with (1, 3, 5, 7, ...) = row 1.
The term "5" is in an odd row, so the 5th Towers of Hanoi move is CW, moving disc #1 (in the first row).
a(3) = 8 in row 4, indicating that the 8th Tower of Hanoi move is CCW, moving disc #4.
A036554 bisects the positive nonzero natural numbers into those in the A036554 set comprising 1/3 of the total numbers, given sufficiently large n.
This corresponds to 1/3 of the TOH moves being CCW and 2/3 CW. Row 1 of the multiplication table = 1/2 of the natural numbers, row 2 = 1/4, row 3 = 1/8 and so on, or 1 = (1/2 + 1/4 + 1/8 + 1/16 + ...). Taking the odd-indexed terms of this series given offset 1, we obtain 2/3 = 1/2 + 1/8 + 1/32 + ..., while sum of the even-indexed terms is 1/3. (End)

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550 (see d(n) on page 501).
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
A. S. Fraenkel, Home Page
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Eric Sopena, i-Mark: A new subtraction division game, arXiv:1509.04199 [cs.DM], 2015.
M. Stoll, Chabauty without the Mordell-Weil group, arXiv preprint arXiv:1506.04286 [math.NT], 2015.
Index entries for 2-automatic sequences.

Indices of odd numbers in A007814. Subsequence of A036552. Complement of A003159. Also double of A003159.

Cf. A000041, A003157, A003158, A005408, A052330, A072939, A079523, A096268 (characteristic function, when interpreted with offset 1), A141290, A174065, A300841.

a036554 = (+ 1) . a079523  -- Reinhard Zumkeller, Mar 01 2012

[2*m:m in [1..100] | Valuation(m,2) mod 2 eq 0]; // Marius A. Burtea, Aug 29 2019

Select[Range[200],OddQ[IntegerExponent[#,2]]&] (* Harvey P. Dale, Oct 19 2011 *)

is(n)=valuation(n,2)%2 \\ Charles R Greathouse IV, Nov 20 2012

def ok(n):
  c = 0
  while n%2 == 0: n //= 2; c += 1
  return c%2 == 1
print([m for m in range(1, 175) if ok(m)]) # Michael S. Branicky, Feb 06 2021

from itertools import count, islice
def A036554_gen(startvalue=1): return filter(lambda n:(~n & n-1).bit_length()&1,count(max(startvalue,1))) # generator of terms >= startvalue
A036554_list = list(islice(A036554_gen(),30)) # Chai Wah Wu, Jul 05 2022

is_A036554 = lambda n: A001511(n)&1==0 # M. F. Hasler, Nov 26 2024

def A036554(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 29 2025

a(n) = A079523(n)+1 = A072939(n)-1.
a(n) = A003156(n) + n = A003157(n) - n = A003158(n) - n + 1. - Philippe Deléham, Apr 10 2004
Values of k such that A091297(k) = 2. - Philippe Deléham, Feb 25 2004
a(n) ~ 3n. - Charles R Greathouse IV, Nov 20 2012 [In fact, a(n) = 3n + O(log n). - Charles R Greathouse IV, Nov 27 2024]
a(n) = 2*A003159(n). - Clark Kimberling, Sep 30 2014
{a(n)} = A052330({A005408(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Aug 26 2019

Incorrect equation removed from formula by Peter Munn, Dec 04 2020

A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 63, 126, 189, 378, 252, 504, 756, 1512, 315, 630, 945, 1890

Offset: 0

Christian G. Bower, Dec 15 1999

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.
Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005
4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005
Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011
The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018
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). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019
The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

			Terms following 5 are 10, 15, 30, 20, 40, 60, 120; this is followed by 7 as 6 has already occurred. - _Philippe Deléham_, Jun 03 2015
From _Antti Karttunen_, Apr 13 2018, after also _Philippe Deléham_'s Jun 03 2015 example: (Start)
This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A300841(k), and each right hand child contains 2*A300841(k), when their parent contains k:
                                     1
                                     |
                  ...................2...................
                 3                                       6
       4......../ \........8                  12......../ \........24
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   5       10         15       30         20       40         60      120
  7 14   21  42     28  56   84  168    35  70  105  210   140 280  420 840
  etc.
Compare also to trees like A005940 and A283477, and sequences A207901 and A302783.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191 (Terms 0..1023 from T. D. Noe)
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..16383, with a color function representing primes in red, perfect powers of primes in gold, squarefree composites in greens, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers. This is a 15 level version of Karttunen's diagram in the example.
OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"
Index entries for sequences that are permutations of the natural numbers

Subsequences: A019565 (squarefree terms), A050376 (the left edge from 2 onward), A336882 (odd terms).
Cf. A050030, A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.
Cf. A000120, A029931, A048793, A064547, A070939, A213925, A299755, A299757, A327041.

a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)

up_to_e = 13; \\ Good for computing up to n = (2^13)-1
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));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); }; \\ Antti Karttunen, Apr 12 2018

a(0)=1; a(n+2^k)=a(n)*b(k) for n < 2^k, k = 0, 1, ... where b is A050376. - Thomas Ordowski, Mar 04 2005
The binary representation of n, n = Sum_{i=0..1+floor(log_2(n))} n_i * 2^i, n_i in {0,1}, is taken as the "Fermi-Dirac representation" (A182979) of a(n), a(n) = Product_{i=0..1+floor(log_2(n))} (b_i)^(n_i) where b_i is A050376(i), i.e., the i-th "Fermi-Dirac prime" (prime power with exponent being a power of 2). - Daniel Forgues, Feb 12 2011
From Antti Karttunen, Apr 12 & 17 2018: (Start)
a(0) = 1; a(2n) = A300841(a(n)), a(2n+1) = 2*A300841(a(n)).
a(n) = A207901(A006068(n)) = A302783(A003188(n)) = A302781(A302845(n)).
(End)

Entry revised Mar 17 2005 by N. J. A. Sloane, based on comments from several people, especially David Wasserman and Thomas Ordowski

A091732 Iphi(n): infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 3, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 6, 24, 12, 16, 18, 28, 8, 30, 15, 20, 16, 24, 24, 36, 18, 24, 12, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 16, 40, 18, 36, 28, 58, 24, 60, 30, 48, 45, 48, 20, 66, 48, 44, 24, 70, 24, 72, 36, 48

Offset: 1

Steven Finch, Mar 05 2004

Not the same as A064380.

With n having a unique factorization as A050376(i) * A050376(j) * ... * A050376(k), with i, j, ..., k all distinct, a(n) = (A050376(i)-1) * (A050376(j)-1) * ... * (A050376(k)-1). (Cf. the first formula). - Antti Karttunen, Jan 15 2019

			a(6)=2 since 6=P_1*P_2, where P_1=2^(2^0) and P_2=3^(2^0); hence (P_1-1)*(P_2-1)=2.
12=3*4 (3,4 are in A050376). Therefore, a(12) = 12*(1-1/3)*(1-1/4) = 6. - _Vladimir Shevelev_, Feb 20 2011

Antti Karttunen, Table of n, a(n) for n = 1..21011
Graeme L. Cohen and Peter Hagis, Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (1993) 373-383.
Steven R. Finch, Unitarism and infinitarism, 2004.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A037445, A049417, A050376, A064380, A300841, A323413.

A091732 := proc(n) local f,a,e,p,b; a :=1 ; for f in ifactors(n)[2] do e := op(2,f) ; p := op(1,f) ; b := convert(e,base,2) ; for i from 1 to nops(b) do if op(i,b) > 0 then a := a*(p^(2^(i-1))-1) ; end if; end do: end do: a ; end proc:
seq(A091732(n),n=1..20) ; # R. J. Mathar, Apr 11 2011

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); Array[a, 100] (* Amiram Eldar, Feb 28 2020 *)

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); }; \\ Antti Karttunen, Jan 15 2019

Consider the set, I, of integers of the form p^(2^j), where p is any prime and j >= 0. Let n > 1. From the fundamental theorem of arithmetic and the fact that the binary representation of any integer is unique, it follows that n can be uniquely factored as a product of distinct elements of I. If n = P_1*P_2*...*P_t, where each P_j is in I, then iphi(n) = Product_{j=1..t} (P_j - 1).
From Vladimir Shevelev, Feb 20 2011: (Start)
Thus we have the following analog of the formula phi(n) = n*Product_{p prime divisors of n} (1-1/p): if the factorization of n over distinct terms of A050376 is n = Product(q) (this factorization is unique), then a(n) = n*Product(1-1/q). Thus a(n) is infinitary multiplicative, i.e., if n_1 and n_2 have no common i-divisors, then a(n_1*n_2) = a(n_1)*a(n_2). Now we see that this property is stronger than the usual multiplicativity, therefore a(n) is a multiplicative arithmetic function.
Add that Sum_{d runs i-divisors of n} a(d)=n and a(n) = n*Sum_{d runs i-divisors of n} A064179(d)/d. The latter formulas are analogs of the corresponding formulas for phi(n): Sum_{d|n} phi(d) = n and phi(n) = n*Sum_{d|n} mu(d)/d. (End).
a(n) = n - A323413(n). - Antti Karttunen, Jan 15 2019
a(n) <= A064380(n), with equality if and only if n is in A050376. - Amiram Eldar, Feb 18 2023

A300840 Fermi-Dirac factorization prime shift towards smaller terms: a(n) = A052330(floor(A052331(n)/2)).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 3, 7, 4, 9, 6, 11, 5, 8, 13, 16, 7, 17, 12, 10, 9, 19, 6, 23, 11, 14, 15, 25, 8, 29, 13, 18, 16, 20, 21, 31, 17, 22, 12, 37, 10, 41, 27, 28, 19, 43, 26, 47, 23, 32, 33, 49, 14, 36, 15, 34, 25, 53, 24, 59, 29, 35, 39, 44, 18, 61, 48, 38, 20, 67, 21, 71, 31, 46, 51, 45, 22, 73, 52, 79, 37, 81, 30, 64, 41, 50, 27

Offset: 1

Antti Karttunen, Apr 13 2018

With n having a unique factorization as fdp(i) * fdp(j) * ... * fdp(k), with i, j, ..., k all distinct, a(n) = fdp(i-1) * fdp(j-1) * ... * fdp(k-1), where fdp(0) = 1 and fdp(n) = A050376(n) for n >= 1.

Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - Andrew Howroyd, Aug 02 2018

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

A left inverse of A300841.
Cf. A050376, A052330, A052331, A059897.
Cf. also A064989.

fdPrimeQ[n_] := Module[{f = FactorInteger[n], e}, Length[f] == 1 && (2^IntegerExponent[(e = f[[1, 2]]), 2] == e)];
prevFDPrime[n_] := Module[{k = n - 1}, While[! fdPrimeQ[k], k--]; k];
fd[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Table[If[b[[j]] > 0, p^(2^(m - j)), Nothing], {j, 1, m}]];
a[n_] := Times @@ prevFDPrime /@ Flatten[fd @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

up_to_e = 8192;
v050376 = vector(up_to_e);
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_e,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); };
A300840(n) = A052330(A052331(n)>>1);

a(n) = A052330(floor(A052331(n)/2)).
For all n >= 1, a(A300841(n)) = n.
a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Nov 30 2019

A305421 GF(2)[X] factorization prime shift towards larger terms.

Original entry on oeis.org

1, 3, 7, 5, 21, 9, 11, 15, 49, 63, 13, 27, 19, 29, 107, 17, 273, 83, 25, 65, 69, 23, 121, 45, 31, 53, 151, 39, 35, 189, 37, 51, 251, 819, 173, 245, 41, 43, 233, 195, 47, 207, 93, 57, 997, 139, 55, 119, 127, 33, 1911, 95, 79, 441, 59, 105, 367, 101, 61, 455, 67, 111, 475, 85, 1281, 269, 73, 1365, 81, 503, 457, 287, 87, 123, 1549, 125, 179, 315

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

Permutation of the odd numbers, A005408.

Let a x b stand for the carryless binary multiplication of positive integers a and b, that is, the result of operation A048720(a,b). With n having a unique factorization as A014580(i) x A014580(j) x ... x A014580(k), 1 <= i <= j <= ... <= k, a(n) = A014580(1+i) x A014580(1+j) x ... x A014580(1+k).

			For n = 12, which by its binary representation '1100' corresponds with (0,1)-polynomial x^3 + x^2, which over GF(2)[X] is factored as (x)(x)(x+1), i.e., 12 = A048720(2,A048720(2,3)) = A048720(A014580(1), A048720(A014580(1),A014580(2))), we then form a(12) as A048720(A014580(2), A048720(A014580(2),A014580(3))) = A048720(3,A048720(3,7)) = 27. Note that x, x+1 and x^2 + x + 1 are the three smallest irreducible (0,1)-polynomials when factored over GF(2)[X], and their binary representations 2, 3 and 7 are the three initial terms of A014580.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A305422 (a left inverse).
Cf. A001317, A014580, A091225, A193231, A268389, A305420, A305423.
Cf. also A003961, A300841.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305420(n) = { my(k=1+n); while(!A091225(k),k++); (k); };
A305421(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305420(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };

For all n >= 1:
A305422(a(n)) = n.
A268389(a(n)) = A007814(n).
a(A000079(n)) = A001317(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

A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2*A300841(n)) = 2*a(n), a(A300841(n)) = A003961(a(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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A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

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A036554 Numbers whose binary representation ends in an odd number of zeros.

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Magma

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A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

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A091732 Iphi(n): infinitary analog of Euler's phi function.

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A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2A300841(n)) = 2a(n), a(A300841(n)) = A003961(a(n)).