|
%I #192 Jul 09 2023 00:01:52
%S 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,
%T 73,79,81,83,89,97,101,103,107,109,113,121,127,131,137,139,149,151,
%U 157,163,167,169,173,179,181,191,193,197,199,211,223,227,229,233,239,241
%N "Fermi-Dirac primes": numbers of the form p^(2^k) where p is prime and k >= 0.
%C 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).
%C 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
%C Except for the first term, same as A084400. - _David Wasserman_, Dec 22 2004
%C The least number having 2^n divisors (=A037992(n)) is the product of the first n terms of this sequence according to Ramanujan.
%C 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
%C 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
%C 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
%C {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
%C 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
%C The first k terms of the sequence solve the following optimization problem:
%C 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
%C From _Joerg Arndt_, Mar 11 2013: (Start)
%C 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).
%C 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)
%C 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
%C Numbers n for which A064547(n) = 1. - _Antti Karttunen_, Feb 10 2016
%C 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
%C 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
%D 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)).
%D S. Ramanujan, Highly Composite Numbers, Collected Papers of Srinivasa Ramanujan, p. 125, Ed. G. H. Hardy et al., AMS Chelsea 2000.
%D 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).
%D J. K. Uhlmann, Dynamic map building and localization: new theoretical foundations, Doctoral Dissertation, University of Oxford, Appendix 16, 1995.
%H T. D. Noe and Charles R Greathouse IV, <a href="/A050376/b050376.txt">Table of n, a(n) for n = 1..10000</a>
%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]
%H Simon Litsyn and Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial CDNumber Theory, 7 (2007), #A33, 1-36.
%H OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>.
%H Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa126-3-1">Compact integers and factorials</a>, Acta Arith. 126 (2007), no.3, 195-236.
%H J. K. Uhlmann, <a href="https://web.archive.org/web/20221103143720/http://vigir.missouri.edu/Papers/JeffUhlmann/Dissertation-p243.pdf">Appendix 16</a>, Doctoral Dissertation, University of Oxford, page 243, 1995.
%F From _Vladimir Shevelev_, Mar 16 2012: (Start)
%F 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) + ...
%F Denote by A(x) the number of terms not exceeding x.
%F Then A(x) = pi(x) + pi(x^(1/2)) + pi(x^(1/4)) + pi(x^(1/8)) + ...
%F Conversely, pi(x) = A(x) - A(sqrt(x)). For example, pi(37) = A(37) - A(6) = 16-4 = 12.
%F (End)
%F A209229(A100995(a(n))) = 1. - _Reinhard Zumkeller_, Mar 19 2013
%F From _Vladimir Shevelev_, Aug 31 2013: (Start)
%F A Fermi-Dirac analog of Euler product: Zeta(s) = Product_{k>=1} (1+a(k)^(-s)), for s > 1.
%F In particular, Product_{k>=1} (1+a(k)^(-2)) = Pi^2/6. (End)
%F a(n) = A268385(A268392(n)). [By their definitions.] - _Antti Karttunen_, Feb 10 2016
%F A000040 union A001248 union A030514 union A179645 union A030635 union .... - _R. J. Mathar_, May 26 2017
%e Prime powers which are not terms of this sequence:
%e 8 = 2^3 = 2^(1+2), 27 = 3^3 = 3^(1+2), 32 = 2^5 = 2^(1+4),
%e 64 = 2^6 = 2^(2+4), 125 = 5^3 = 5^(1+2), 128 = 2^7 = 2^(1+2+4)
%e "Fermi-Dirac factorizations":
%e 6 = 2*3, 8 = 2*4, 24 = 2*3*4, 27 = 3*9, 32 = 2*16, 64 = 4*16,
%e 108 = 3*4*9, 120 = 2*3*4*5, 121 = 121, 125 = 5*25, 128 = 2*4*16.
%p isA050376 := proc(n)
%p local f,e;
%p f := ifactors(n)[2] ;
%p if nops(f) = 1 then
%p e := op(2,op(1,f)) ;
%p if isA000079(e) then
%p true;
%p else
%p false;
%p end if;
%p else
%p false;
%p end if;
%p end proc:
%p A050376 := proc(n)
%p option remember ;
%p local a;
%p if n = 1 then
%p 2 ;
%p else
%p for a from procname(n-1)+1 do
%p if isA050376(a) then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc: # _R. J. Mathar_, May 26 2017
%t 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 *)
%o (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
%o (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
%o (PARI) is_A050376(n)=2^#binary(n=isprimepower(n))==n*2 \\ _M. F. Hasler_, Apr 08 2015
%o (PARI) ispow2(n)=n && n>>valuation(n,2)==1
%o is(n)=ispow2(isprimepower(n)) \\ _Charles R Greathouse IV_, Sep 18 2015
%o (Haskell)
%o a050376 n = a050376_list !! (n-1)
%o a050376_list = filter ((== 1) . a209229 . a100995) [1..]
%o -- _Reinhard Zumkeller_, Mar 19 2013
%o (Scheme)
%o (define A050376 (MATCHING-POS 1 1 (lambda (n) (= 1 (A064547 n)))))
%o ;; Requires also my IntSeq-library. - _Antti Karttunen_, Feb 09 2016
%o (Python)
%o from sympy import isprime, perfect_power
%o def ok(n):
%o if isprime(n): return True
%o answer = perfect_power(n)
%o if not answer: return False
%o b, e = answer
%o if not isprime(b): return False
%o while e%2 == 0: e //= 2
%o return e == 1
%o def aupto(limit):
%o alst, m = [], 1
%o for m in range(1, limit+1):
%o if ok(m): alst.append(m)
%o return alst
%o print(aupto(241)) # _Michael S. Branicky_, Feb 03 2021
%Y Cf. A000040 (primes, is a subsequence), A026416, A026477, A050377-A050380, A052330, A064547, A066724, A084400, A176699, A182979.
%Y Cf. A268388 (complement without 1).
%Y Cf. A124010, subsequence of A000028, A000961, A213925, A223490.
%Y Cf. A228520, A186945 (Fermi-Dirac analog of Ramanujan primes, A104272, and Labos primes, A080359).
%Y Cf. also A268385, A268391, A268392.
%K nonn,easy,nice
%O 1,1
%A _Christian G. Bower_, Nov 15 1999
%E Edited by _Charles R Greathouse IV_, Mar 17 2010
%E More examples from _Daniel Forgues_, Feb 09 2011
| 