cp's OEIS Frontend

A090885 Sum of the squares of the exponents in the prime factorization of n.

%I A090885 #19 Mar 21 2023 15:15:02
%S A090885 0,1,1,4,1,2,1,9,4,2,1,5,1,2,2,16,1,5,1,5,2,2,1,10,4,2,9,5,1,3,1,25,2,
%T A090885 2,2,8,1,2,2,10,1,3,1,5,5,2,1,17,4,5,2,5,1,10,2,10,2,2,1,6,1,2,5,36,2,
%U A090885 3,1,5,2,3,1,13,1,2,5,5,2,3,1,17,16,2,1,6,2,2,2,10,1,6,2,5,2,2,2,26,1,5,5,8
%N A090885 Sum of the squares of the exponents in the prime factorization of n.
%C A090885 From _Daniel Forgues_, Mar 30 2009: (Start)
%C A090885 Euclidean norm (square of the length as measured from the origin 0 which represents the number 1) of the exponents vector of n.
%C A090885 If we consider n as represented as an exponents vector in an infinite dimensional discrete vector space (infinite dimensional lattice) where each dimension corresponds to a prime {p1, p2, p3, p4, p5, p6, ...} = {2, 3, 5, 7, 11, 13, ...} then the product of n1 with n2 corresponds to vector addition of the exponents vectors of n1 and n2.
%C A090885 If 2 numbers n1 and n2 are coprime then the length of the exponents vector of the product n1*n2 is the Pythagorean sum of the lengths of the exponents vectors of n1 and n2.
%C A090885 For the product of 2 arbitrary numbers n1 and n2 we have the triangle inequality applying to the lengths of the exponents vectors of n1, n2, n1*n2. E.g., 107653 = 7^2 * 13^3 is represented as (0, 0, 0, 2, 0, 3, 0, 0, 0, ...) as an exponents vector in an infinite dimensional space associated with the primes.
%C A090885 If all the coordinates of the exponents vector are positive we have the representation of an integer. If some components are negative then we have the representation of a rational number. The origin 0 corresponds to the number 1. There is no representation for 0 as an exponents vector.
%C A090885 If 2 numbers are coprime then their exponents vectors are orthogonal. If the exponents vectors of 2 numbers n1 and n2 are parallel then we have n1^a = n2^b for some nonzero integers a and b. (End)
%C A090885 Rényi & Turán prove that the Erdős-Kac theorem holds for this sequence: its values are normally distributed with mean and variance log log n, see Theorem 3. - _Charles R Greathouse IV_, Mar 21 2023
%D A090885 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, p. 155.
%H A090885 Daniel Forgues, <a href="/A090885/b090885.txt">Table of n, a(n) for n = 1..100000</a>
%H A090885 R. L. Duncan, <a href="https://www.jstor.org/stable/2312731">A class of additive arithmetical functions</a>, The American Mathematical Monthly, Vol. 69, No. 1 (1962), pp. 34-36.
%H A090885 Alfréd Rényi and Pál Turán, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa417.pdf">On a theorem of Erdös-Kac</a>, Acta Arithmetica 4.1 (1958), pp. 71-84.
%F A090885 Additive with a(p^e) = e^2.
%F A090885 Sum_{k=1..n} a(k) ~ n * log(log(n)) + B_2 * n + O(n/log(n)), where B_2 = gamma + Sum_{p prime} ((1-1/p)*Sum_{m>=1} m^2/p^m + log(1-1/p)), and gamma is Euler's constant (Duncan, 1962). - _Amiram Eldar_, Mar 05 2021
%t A090885 Join[{0},Table[Total[FactorInteger[n][[All,2]]^2],{n,2,100}]] (* _Harvey P. Dale_, Apr 25 2020 *)
%o A090885 (PARI) a(n,f=factor(n))=norml2(f[,2]) \\ _Charles R Greathouse IV_, Mar 09 2021
%Y A090885 Cf. A001222, A001620.
%K A090885 easy,nonn
%O A090885 1,4
%A A090885 _Sam Alexander_, Dec 12 2003
%E A090885 More terms from _Ray Chandler_, Dec 20 2003