cp's OEIS Frontend

A071625 Number of distinct exponents when n is factorized as a product of primes.

%I A071625 #82 Jun 15 2025 09:56:27
%S A071625 0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,1,1,1,1,1,
%T A071625 1,1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,2,1,2,1,2,1,1,1,2,1,1,2,1,1,1,1,2,
%U A071625 1,1,1,2,1,1,2,2,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,2,2,1,1,1,1,2,1
%N A071625 Number of distinct exponents when n is factorized as a product of primes.
%C A071625 First term greater than 2 is a(360) = 3.
%C A071625 From _Michel Marcus_, Apr 24 2016: (Start)
%C A071625 A006939(n) gives the least m such that a(m) = n.
%C A071625 A062770 is the sequence of integers m such that a(m) = 1. (End)
%C A071625 We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first two omegas are A001222 and A001221, while this sequence is the third, and A323022 is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices). - _Gus Wiseman_, Jan 02 2019
%C A071625 Sanna (2020) proved that for each k>=1, the sequence of numbers n with A071625(n) = k has an asymptotic density A_k = (6/Pi^2) * Sum_{n>=1, n squarefree} rho_k(n)/psi(n), where psi is the Dedekind psi function (A001615), and rho_k(n) is defined by rho_1(n) = 1 if n = 1 and 0 otherwise, rho_{k+1}(n) = 0 if n = 1 and (1/(n-1)) * Sum_{d|n, d<n} rho_k(d) otherwise. - _Amiram Eldar_, Oct 18 2020
%H A071625 Giovanni Resta, <a href="/A071625/b071625.txt">Table of n, a(n) for n = 1..10000</a>
%H A071625 E. T. Bell, <a href="http://dx.doi.org/10.1090/S0002-9904-1937-06651-1">Functions of coprime divisors of integers</a>, Bull. Amer. Math. Soc. 43 (1937), 818-822.
%H A071625 Carlo Sanna, <a href="https://doi.org/10.1007/s12044-020-0556-y">On the number of distinct exponents in the prime factorization of an integer</a>, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, <a href="https://www.ias.ac.in/describe/article/pmsc/130/0027">alternative link</a>, <a href="https://arxiv.org/abs/1902.09224">arXiv preprint</a>, arXiv:1902.09224 [math.NT], 2019.
%e A071625 n = 5040 = 2^4*(3*5)^2*7, three different exponents arise:4,2 and 1; so a(5040)=3.
%p A071625 # Using function 'PrimeSignature' from A124010.
%p A071625 a := n -> nops(convert(PrimeSignature(n), set)):
%p A071625 seq(a(n), n = 1..105); # _Peter Luschny_, Jun 15 2025
%t A071625 ffi[x_] := Flatten[FactorInteger[x]];
%t A071625 lf[x_] := Length[FactorInteger[x]];
%t A071625 ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}];
%t A071625 Table[Length[Union[ep[w]]], {w, 1, 256}]
%t A071625 (* Second program: *)
%t A071625 {0}~Join~Array[Length@ Union@ FactorInteger[#][[All, -1]] &, 104, 2] (* _Michael De Vlieger_, Apr 10 2019 *)
%o A071625 (PARI) a(n) = #Set(factor(n)[,2]); \\ _Michel Marcus_, Mar 12 2015
%o A071625 (Python)
%o A071625 from sympy import factorint
%o A071625 def a(n): return len(set(factorint(n).values()))
%o A071625 print([a(n) for n in range(1, 106)]) # _Michael S. Branicky_, Sep 01 2022
%Y A071625 Cf. A051903, A051904, A006939, A124010.
%Y A071625 Cf. A001221, A001222, A001615, A059404, A062770, A118914, A181819, A323014, A323022, A323023, A323024, A323025.
%K A071625 nonn
%O A071625 1,12
%A A071625 _Labos Elemer_, May 29 2002