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A050376
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"Fermi-Dirac primes": numbers of the form p^(2^k) where p is prime and k >= 0.
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245
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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
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OFFSET
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1,1
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COMMENTS
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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
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_1<x_2<...<x_k, A064547(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
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
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
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REFERENCES
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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.
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LINKS
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J. K. Uhlmann, Appendix 16, Doctoral Dissertation, University of Oxford, page 243, 1995.
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FORMULA
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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)
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)
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EXAMPLE
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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.
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MAPLE
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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:
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;
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n)= local(m, c, k, p); if(n<=1, 2*(n==1), n--; c=0; m=2; while( c<n, m++; if( isprime(m) || ( (k=ispower(m, , &p))&&isprime(p)&& (k==2^valuation(k, 2)) ), c++)); m)} \\ Michael 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) ispow2(n)=n && n>>valuation(n, 2)==1
(Haskell)
a050376 n = a050376_list !! (n-1)
a050376_list = filter ((== 1) . a209229 . a100995) [1..]
(Scheme)
(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
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CROSSREFS
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Cf. A000040 (primes, is a subsequence), A026416, A026477, A050377-A050380, A052330, A064547, A066724, A084400, A176699, A182979.
Cf. A268388 (complement without 1).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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