cp's OEIS Frontend

Sequence linked to A050376.

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

Or a(n) is the maximal m for which the Fermi-Dirac representation of m! (see comment in A050376) contains single power of 2 and single power of prime(n).

Or a(n) is the maximal m for which the Fermi-Dirac representation of m! (see comment in A050376) contains single power of 2 and single power of prime(n).

For n=2 the corresponding value is not known; moreover, we do not know if this value is finite (in any case, it is not less than 524306). See also comment to A177458.

If a(2) exists, then it is at least 81129638414606681695789005144146. - Charles R Greathouse IV, Apr 10 2012

For any number k > 1, write its "compact prime factorization", with no spaces, as p1^e1*p2^e2*...*pj^ej, where p1, p2, ..., pj are the distinct prime factors of k and e1, e2, ..., ej are their respective exponents, but omit each exponent whose value is 1 (along with its caret character "^"). Sequence gives those numbers k whose compact prime factorization has fewer characters than k has decimal digits.

The smallest term other than a prime power is 109375 = 5^6*7.

The smallest term that is a power of 10 is 10000000 = 2^7*5^7.

The smallest term that is a factorial is 45!

= 119622220865480194561963161495657715064383733760000000000

= 2^41*3^21*5^10*7^6*11^4*13^3*17^2*19^2*23*29*31*37*41*43.

Includes 2^k for k >= 14, 3^k for k >= 7, 5^k for k >= 5, 7^k for k >= 4. - Robert Israel, Dec 26 2016

Let k'(b) be the smallest k such that b^k is included; then the sequence k'(2), k'(3), k'(4), ... begins {14, 7, 7, 5, 9, 4, 5, 4, 7, 4, 8, 4, 7, 6, 4, 4, 7, 4, 7, 6, 6, 3, 6, 3, ...} (with the larger values generally occurring where b has more than one prime divisor). It appears that b^k is included for all b > 1 and all k >= k'(b) with only two exceptions: although 6^k'(6) = 6^9 = 10077696 = 2^9*3^9 and 6^12 = 2176782336 = 2^12*3^12 are included, 6^10 = 60466176 = 2^10*3^10 and 6^11 = 362797056 = 2^11*3^11 are not. - Jon E. Schoenfield, Dec 26 2016

Note that there is another class of numbers that are called "Compact". See the definition in A169661. See also the links from T. M. Apostol and from V. Shevelev in the same entry. See also A070566 and A145554. - Omar E. Pol, Dec 26 2016