cp's OEIS Frontend

Every number n is a product of a unique subset of these numbers. This is sometimes called the Fermi-Dirac factorization of n (see A182979). Proof: In the prime factorization n = Product_{j>=1} p(j)^e(j) expand every exponent e(j) as binary number and pick the terms of this sequence corresponding to the positions of the ones in binary (it is clear that both n and n^2 have the same number of factors in this sequence, and that each factor appears with exponent 1 or 0).

Or, a(1) = 2; for n>1, a(n) = smallest number which cannot be obtained as the product of previous terms. This is evident from the unique factorization theorem and the fact that every number can be expressed as the sum of powers of 2. - Amarnath Murthy, Jan 09 2002

Except for the first term, same as A084400. - David Wasserman, Dec 22 2004

The least number having 2^n divisors (=A037992(n)) is the product of the first n terms of this sequence according to Ramanujan.

According to the Bose-Einstein distribution of particles, an unlimited number of particles may occupy the same state. On the other hand, according to the Fermi-Dirac distribution, no two particles can occupy the same state (by the Pauli exclusion principle). Unique factorizations of the positive integers by primes (A000040) and over terms of A050376 one can compare with two these distributions in physics of particles. In the correspondence with this, the factorizations over primes one can call "Bose-Einstein factorizations", while the factorizations over distinct terms of A050376 one can call "Fermi-Dirac factorizations". - Vladimir Shevelev, Apr 16 2010

The numbers of the form p^(2^k), where p is prime and k >= 0, might thus be called the "Fermi-Dirac primes", while the classic primes might be called the "Bose-Einstein primes". - Daniel Forgues, Feb 11 2011

In the theory of infinitary divisors, the most natural name of the terms is "infinitary primes" or "i-primes". Indeed, n is in the sequence, if and only if it has only two infinitary divisors. Since 1 and n are always infinitary divisors of n>1, an i-prime has no other infinitary divisors. - Vladimir Shevelev, Feb 28 2011

{a(n)} is the minimal set including all primes and closed with respect to squaring. In connection with this, note that n and n^2 have the same number of factors in their Fermi-Dirac representations. - Vladimir Shevelev, Mar 16 2012

In connection with this sequence, call an integer compact if the factors in its Fermi-Dirac factorization are pairwise coprime. The density of such integers equals (6/Pi^2)*Product_{prime p} (1+(Sum_{i>=1} p^(-(2^i-1))/(p+1))) = 0.872497... It is interesting that there exist only 7 compact factorials listed in A169661. - Vladimir Shevelev, Mar 17 2012

The first k terms of the sequence solve the following optimization problem:

Let x_1, x_2,..., x_k be integers with the restrictions: 2<=x_1A064547(Product{i=1..k} x_i) >= k. Let the goal function be Product_{i=1..k} x_i. Then the minimal value of the goal function is Product_{i=1..k} a(i). - Vladimir Shevelev, Apr 01 2012

From Joerg Arndt, Mar 11 2013: (Start)

Similarly to the first comment, for the sequence "Numbers of the form p^(3^k) or p^(2*3^k) where p is prime and k >= 0" one obtains a factorization into distinct factors by using the ternary expansion of the exponents (here n and n^3 have the same number of such factors).

The generalization to base r would use "Numbers of the form p^(r^k), p^(2*r^k), p^(3*r^k), ..., p^((r-1)*r^k) where p is prime and k >= 0" (here n and n^r have the same number of (distinct) factors). (End)

The first appearance of this sequence as a multiplicative basis in number theory with some new notions, formulas and theorems may have been in my 1981 paper (see the Abramovich reference). - Vladimir Shevelev, Apr 27 2014

Numbers n for which A064547(n) = 1. - Antti Karttunen, Feb 10 2016

Lexicographically earliest sequence of distinct nonnegative integers such that no term is a product of 2 or more distinct terms. Removing the distinctness requirement, the sequence becomes A000040 (the prime numbers); and the equivalent sequence where the product is of 2 distinct terms is A026416 (without its initial term, 1). - Peter Munn, Mar 05 2019

The sequence was independently developed as a multiplicative number system in 1985-1986 (and first published in 1995, see the Uhlmann reference) using a proof method involving representations of positive integers as sums of powers of 2. This approach offers an arguably simpler and more flexible means for analyzing the sequence. - Jeffrey K. Uhlmann, Nov 09 2022

Previous name: sequence consists of products of distinct relatively prime terms of A084400. - Vladimir Shevelev, Sep 24 2015

These numbers are also called "compact integers."

The density of this sequence exists and equals 0.872497...

There exist only seven compact factorials A000142(n) for n=1,2,3,6,7,10 and 11.

For a general definition of exponentially S-numbers, see comments in A209061. - Vladimir Shevelev, Sep 24 2015

The first 1000 digits of the density of the sequence were calculated by Juan Arias-de-Reyna in A271727. - Vladimir Shevelev, Apr 18 2016

A225546 maps the set of terms 1:1 onto A268375. - Peter Munn, Jan 26 2020

Numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide. - Amiram Eldar, Dec 23 2020

This sequence and A000379 (its complement) give the unique solution to the problem of splitting the positive integers into two classes in such a way that products of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000069, A001969.

Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007

The sequence contains products of odd number of distinct terms of A050376. - Vladimir Shevelev, May 04 2010

From Vladimir Shevelev, Oct 28 2013: (Start)

Numbers m such that infinitary Moebius function of m (A064179) equals -1. This follows from the definition of A064179.

Number m is in the sequence if and only if the number k = k(m) of terms of A050376 which divide m with odd maximal exponent is odd.

For example, if m = 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 k(96) = 3 and 96 is a term.

(End)

Positions of odd terms in A064547, A268386 and A293439. - Antti Karttunen, Nov 09 2017

Lexicographically earliest sequence of distinct nonnegative integers such that no term is the A059897 product of 2 terms. (A059897 can be considered as a multiplicative operator related to the Fermi-Dirac factorization of numbers described in A050376.) Specifying that the A059897 product be of 2 distinct terms leaves the sequence unchanged. The equivalent sequences using standard integer multiplication are A026416 (with the 2 terms specified as distinct) and A026424 (otherwise). - Peter Munn, Mar 16 2019

From Amiram Eldar, Oct 02 2024: (Start)

Numbers whose number of infinitary divisors (A037445) is not a square.

Numbers whose exponentially odious part (A367514) has an odd number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 1. (End)

From Bob Selcoe, Aug 25 2016: (Start)

Once a term with a given prime signature S (i.e., multiset of prime exponents) appears, then all numbers with the same prime signature follow. So either all or no terms with the same prime signature appear (first conjectured by Charles R Greathouse IV).

Proof:

i. Let S = {i,j,k..,y}, i>=j>=k.. be prime signatures, i.e., numbers of the form (p^i*q^j*r^k..*z^y) where {p,q,r,..,z} are distinct primes; denote S = {} as the signature for p^0 = 1.

ii. By definition, 1 and all primes p appear in the sequence; so terms where S = {k} (i.e., one-digit signatures denoting prime powers p^k) are k=0 and k = A026474(n) = {1,2,4,8,15,22,29..}, because k cannot be the sum of any combination of 3 smaller k. So the only prime power terms are p^0 = 1 and p^A026474(n), or S = {}, {1}, {2}, {4}, {8}, {15}, {22}, {29}...

iii. By induction, once S = {k} is determined, all other terms with the same prime signature appear (or do not) depending on the various combinations of signature exponents of previous terms with smaller signature sums. So for example, terms with the same two-digit signature S = {k,k} (i.e. (pq)^k) are constrained by the following: since p = {1} and q = {1} appear, then (pq) = {1,1} does not because in this case {1}*{1}*{} = {1,1}; since p^2 = {2} and q^2 = {2} appear, then (pq)^2 = {2,2} does not; since {4} appears but {3} does not, then {3,3} appears but {4,4} and {5,5} do not (the latter because {3,3}*{2}*{2} = {5,5} when p^2,q^2 = {2}). Note that {3,3} would not appear if {3,2} appeared because {3,2}*{1}*{} = {3,3} when q = {1}; but because p = {1} and p^2,q^2 = {2} appear, then {2}*{2}*{1} = {3,2} does not. {6,6} does not appear because {6,5} appears (by virtue of other constraints) and {6,5}*{1}*{} = {6,6} when q = {1}. Determining which signatures appear and which do not becomes increasingly complicated as the sequence increases.

Don Reble offered a proof on the Sequence Fans Mailing List which seems to be different (and certainly more formal) than mine. Perhaps mine is more of an "explanation" than a "proof"? (End)

An equivalent definition is: a(1) = 1, a(2) = 2; and for n > 2, a(n) = least positive integer > a(n-1) and not of the form a(i)*a(j) for 1 <= i < j < n.

a(2) to a(29) match the initial terms of A000028. [corrected by Peter Munn, Mar 15 2019]

This has a simpler definition than A000028, but the resulting pair lacks the crucial property of the A000028/A000379 pair (see the comment in A000028). - N. J. A. Sloane, Sep 28 2007

Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007

From Vladimir Shevelev, Apr 05 2013: (Start)

1) The sequence does not contain (for example) 140, so is different from A000028.

2) Representation of numbers which are absent in the sequence as a product of two different terms of the sequence is, generally speaking, not unique. For example, 210 = 2*105 = 3*70 = 5*42 = 7*30.

(End)

Excluding a(1) = 1, the lexicographically earliest sequence of distinct nonnegative integers such that no term is a product of 2 distinct terms. Removing the latter distinctness requirement, the sequence becomes A026424; and the equivalent sequence where the product is of 2 or more distinct terms is A050376. A000028 is similarly the equivalent sequence when A059897 is used as multiplicative operator in place of standard integer multiplication. - Peter Munn, Mar 15 2019

Daniel Forgues (see A182979) and Reinhard Zumkeller (see A213925) describe the increasing sequence of positive integers of the form p^(2^k) where p is prime and k>=0 (A050376 or A084400) as Fermi-Dirac primes, because any positive integer has a unique factorization into distinct terms.

The first 15 terms are the same as A026477; the first 13 terms are the same as A026416.

Contains all primes. - Ivan Neretin, Mar 02 2016

A Fermi-Dirac prime (A050376) is a positive integer of the form p^(2^k) where p is prime and k>=0.

In analogy with the Matula-Goebel correspondence between rooted trees and positive integers (see A061775), the iterated normalized Fermi-Dirac representation gives a correspondence between rooted identity trees and positive integers. Then a(n) is the number of nodes in the rooted identity tree corresponding to n.

Ordinal transform of A223491, or equally, of A302785.

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