cp's OEIS Frontend

A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

%I A212177 #26 Jul 23 2024 10:51:19
%S A212177 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,
%T A212177 1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A212177 1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,1,1,2,1,1
%N A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).
%C A212177 Length of second signature of A013929(n) (cf. A212172).
%D A212177 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
%H A212177 Reinhard Zumkeller, <a href="/A212177/b212177.txt">Table of n, a(n) for n = 1..10000</a>
%H A212177 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A212177 Primefan, <a href="http://primefan.tripod.com/500factored.html">The First 2500 Integers Factored</a> (1st of 5 pages).
%F A212177 a(n) = A056170(A013929(n)).
%F A212177 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (Sum_{p prime} 1/p^2)/(1-1/zeta(2)) = A085548 / A229099 = 1.15347789194214704903... . - _Amiram Eldar_, Oct 01 2023
%e A212177 24 = 2^3*3 has 1 exponent of size 2 or greater in its prime factorization. Since 24 = A013929(8), a(8) = 1.
%t A212177 f[n_] := Module[{c = Count[FactorInteger[n][[;; , 2]], _?(# > 1&)]}, If[n > 1 && c > 0, c, Nothing]]; f[1] = 0; Array[f, 300] (* _Amiram Eldar_, Oct 01 2023 *)
%o A212177 (Haskell)
%o A212177 a212177 n = a212177_list !! (n-1)
%o A212177 a212177_list = filter (> 0) a056170_list
%o A212177 -- _Reinhard Zumkeller_, Dec 29 2012
%o A212177 (Python)
%o A212177 from math import isqrt
%o A212177 from sympy import mobius, factorint
%o A212177 def A212177(n):
%o A212177     def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A212177     m, k = n, f(n)
%o A212177     while m != k:
%o A212177         m, k = k, f(k)
%o A212177     return sum(1 for e in factorint(m).values() if e>1) # _Chai Wah Wu_, Jul 22 2024
%Y A212177 Cf. A013929, A212172, A212174.
%Y A212177 Cf. A046028, A056170, A085548, A124010, A229099.
%K A212177 nonn,easy
%O A212177 1,13
%A A212177 _Matthew Vandermast_, Jun 03 2012