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

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, 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, 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

Offset: 1

Author

Sam Alexander, Dec 12 2003

Keywords

• easy
• nonn

Comments

From Daniel Forgues, Mar 30 2009: (Start)

Euclidean norm (square of the length as measured from the origin 0 which represents the number 1) of the exponents vector of n.

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.

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.

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.

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.

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)

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

References

• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, p. 155.

Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000
• R. L. Duncan, A class of additive arithmetical functions, The American Mathematical Monthly, Vol. 69, No. 1 (1962), pp. 34-36.
• Alfréd Rényi and Pál Turán, On a theorem of Erdös-Kac, Acta Arithmetica 4.1 (1958), pp. 71-84.

Crossrefs

Cf. A001222, A001620.

Programs

Mathematica

Join[{0},Table[Total[FactorInteger[n][[All,2]]^2],{n,2,100}]] (* Harvey P. Dale, Apr 25 2020 *)

PARI

a(n,f=factor(n))=norml2(f[,2]) \\ Charles R Greathouse IV, Mar 09 2021

Formula

Additive with a(p^e) = e^2.

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

Extensions

More terms from Ray Chandler, Dec 20 2003