login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A050376 Numbers of the form p^(2^k) where p is prime and k >= 0. 108
2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Every number n is a product of a unique subset of these numbers: in the prime factorization n = prod( 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 can not 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 (it is 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 is closed with respect to  operation of the raising to power 2. 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, it is introduced a notion "compact integer". An integer is called compact, if the factors in its Fermi-Dirac factorization are pairwise coprime. The density of such integers equals (6/pi^2)*prod{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_1<x_2<...<x_k, A064547(prod{i=1,...,k}x_i)>=k.  Let the goal function be prod{i=1,...,k}x_i. Then  minimal value of the goal function is  prod{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

REFERENCES

V. S. Abramovich, On an analog of the Euler function, Proceeding of the North-Caucasus Center of the Academy of Sciences of the USSR (Rostov na Donu) (1981) No. 2, 13-17 (Russian; MR0632989(83a:10003)).

S. Ramanujan, Highly Composite Numbers, Collected Papers of Srinivasa Ramanujan, p. 125, Ed. G. H. Hardy et al., AMS Chelsea 2000.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

S. R. Finch, Unitarism and infinitarism.

Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

OEIS Wiki, "Fermi-Dirac representation" of n.

V. Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no.3, 195-236.

FORMULA

From Vladimir Shevelev, Mar 16 2012 (Start)

Prod_(i>=1) a(i)^k_i=n!, where k_i=floor(n/a(i))-floor(n/a(i)^2)+floor(n/a(i)^3)-floor(n/a(i)^4)+...

Denote by A(x) the number of terms not exceeding x.

Then A(x)=pi(x)+pi(x^(1/2))+pi(x^(1/4))+pi(x^(1/8))+...

Conversely, pi(x)=A(x)-A(sqrt(x)). For example, pi(37)=A(37)-A(6)=16-4=12.

(End)

A209229(A100995(n)) = 1. - Reinhard Zumkeller, Mar 19 2013

A Fermi-Dirac analog of Euler product: for s>1, Zeta(s) = prod{k>=1}(1+a(k)^(-s)).

In particular, prod{k>=1}(1+a(k)^(-2)) = pi^2/6. - Vladimir Shevelev, Aug 31 2013

EXAMPLE

Prime powers which are not members of this sequence:

  8 = 2^3 = 2^(1+2), 27 = 3^3 = 3^(1+2), 32 = 2^5 = 2^(1+4),

  64 = 2^6 = 2^(2+4), 125 = 5^3 = 5^(1+2), 128 = 2^7 = 2^(1+2+4)

"Fermi-Dirac factorizations":

  6 = 2*3, 8 = 2*4, 24 = 2*3*4, 27 = 3*9, 32 = 2*16, 64 = 4*16,

  108 = 3*4*9, 120 = 2*3*4*5, 121 = 121, 125 = 5*25, 128 = 2*4*16.

MATHEMATICA

nn = 300; t = {}; k = 1; While[lim = nn^(1/k); lim > 2,  t = Join[t, Prime[Range[PrimePi[lim]]]^k]; k = 2 k]; t = Union[t] (* T. D. Noe, Apr 05 2012 *)

PROG

(PARI) {a(n)= local(m, c, k, p); if(n<=0, 2*(n==0), c=0; m=2; while( c<n, m++; if( isprime(m)| ( (k=ispower(m, (null), &p))&isprime(p)& k ==2^valuation(k, 2) ), c++)); m)} /* Michael Somos, Apr 15 2005 */

(PARI) lst(lim)=my(v=primes(primepi(lim)), t); forprime(p=2, sqrt(lim), t=p; while((t=t^2)<=lim, v=concat(v, t))); vecsort(v) \\ Charles R Greathouse IV, Apr 10 2012

(Haskell)

a050376 n = a050376_list !! (n-1)

a050376_list = filter ((== 1) . a209229 . a100995) [1..]

-- Reinhard Zumkeller, Mar 19 2013

CROSSREFS

Cf. A000028, A000040 (primes), A026416, A026477, A050377-A050380, A052330, A066724, A084400, A176699, A182979.

Cf. A124010, subsequence of A000961, A213925, A223490, A223490.

Cf. A228520, A186945 (Fermi-Dirac analog of Ramanujan primes, A104272, and Labos primes, A080359).

Sequence in context: A026477 A079852 A084400 * A050198 A158923 A008740

Adjacent sequences:  A050373 A050374 A050375 * A050377 A050378 A050379

KEYWORD

nonn,easy,nice

AUTHOR

Christian G. Bower, Nov 15 1999.

EXTENSIONS

Edited by Charles R Greathouse IV, Mar 17 2010

More examples from Daniel Forgues, Feb 09 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 20 17:47 EST 2014. Contains 252272 sequences.