cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A050376 "Fermi-Dirac primes": numbers of the form p^(2^k) where p is prime and k >= 0.

Original entry on oeis.org

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
Offset: 1

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Author

Christian G. Bower, Nov 15 1999

Keywords

Comments

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

Examples

			Prime powers which are not terms 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.

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).
  • J. K. Uhlmann, Dynamic map building and localization: new theoretical foundations, Doctoral Dissertation, University of Oxford, Appendix 16, 1995.

Links

Crossrefs

Cf. A000040 (primes, is a subsequence), A026416, A026477, A037992 (partial products), A050377-A050380, A052330, A064547, A066724, A084400, A176699, A182979.
Cf. A268388 (complement without 1).
Cf. A124010, subsequence of A000028, A000961, A213925, A223490.
Cf. A228520, A186945 (Fermi-Dirac analog of Ramanujan primes, A104272, and Labos primes, A080359).
Cf. also A268385, A268391, A268392.

Programs

  • Haskell
    a050376 n = a050376_list !! (n-1)
a050376_list = filter ((== 1) . a209229 . a100995) [1..]
-- Reinhard Zumkeller, Mar 19 2013
  • Maple
    isA050376 := proc(n)
    local f,e;
    f := ifactors(n)[2] ;
    if nops(f) = 1 then
        e := op(2,op(1,f)) ;
        if isA000079(e) then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
A050376 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2 ;
    else
        for a from procname(n-1)+1 do
            if isA050376(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 26 2017
  • 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 *)
  • PARI
    {a(n)= local(m, c, k, p); if(n<=1, 2*(n==1), n--; c=0; m=2; while( cMichael Somos, Apr 15 2005; edited by Michel Marcus, Aug 07 2021
  • 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
  • PARI
    is_A050376(n)=2^#binary(n=isprimepower(n))==n*2 \\ M. F. Hasler, Apr 08 2015
  • PARI
    ispow2(n)=n && n>>valuation(n,2)==1
is(n)=ispow2(isprimepower(n)) \\ Charles R Greathouse IV, Sep 18 2015
  • PARI
    isok(n)={my(e=isprimepower(n)); e && !bitand(e,e-1)} \\ Andrew Howroyd, Oct 16 2024
  • Python
    from sympy import isprime, perfect_power
def ok(n):
  if isprime(n): return True
  answer = perfect_power(n)
  if not answer: return False
  b, e = answer
  if not isprime(b): return False
  while e%2 == 0: e //= 2
  return e == 1
def aupto(limit):
  alst, m = [], 1
  for m in range(1, limit+1):
    if ok(m): alst.append(m)
  return alst
print(aupto(241)) # Michael S. Branicky, Feb 03 2021
  • Python
    from sympy import primepi, integer_nthroot
def A050376(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): return n+x-sum(primepi(integer_nthroot(x,1<Chai Wah Wu, Feb 18-19 2025
  • Scheme
    (define A050376 (MATCHING-POS 1 1 (lambda (n) (= 1 (A064547 n)))))
;; Requires also my IntSeq-library. - Antti Karttunen, Feb 09 2016

Formula

From Vladimir Shevelev, Mar 16 2012: (Start)
Product_{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(a(n))) = 1. - Reinhard Zumkeller, Mar 19 2013
From Vladimir Shevelev, Aug 31 2013: (Start)
A Fermi-Dirac analog of Euler product: Zeta(s) = Product_{k>=1} (1+a(k)^(-s)), for s > 1.
In particular, Product_{k>=1} (1+a(k)^(-2)) = Pi^2/6. (End)
a(n) = A268385(A268392(n)). [By their definitions.] - Antti Karttunen, Feb 10 2016
A000040 union A001248 union A030514 union A179645 union A030635 union .... - R. J. Mathar, May 26 2017

Extensions

Edited by Charles R Greathouse IV, Mar 17 2010
More examples from Daniel Forgues, Feb 09 2011

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