cp's OEIS Frontend

A285329 a(n) = A013928(A007947(n)).

%I A285329 #24 May 12 2024 02:01:16
%S A285329 0,1,2,1,3,4,5,1,2,6,7,4,8,9,10,1,11,4,12,6,13,14,15,4,3,16,2,9,17,18,
%T A285329 19,1,20,21,22,4,23,24,25,6,26,27,28,14,10,29,30,4,5,6,31,16,32,4,33,
%U A285329 9,34,35,36,18,37,38,13,1,39,40,41,21,42,43,44,4,45,46,10,24,47,48,49,6,2,50,51,27,52,53,54,14,55,18,56,29,57,58,59,4,60,9,20,6
%N A285329 a(n) = A013928(A007947(n)).
%C A285329 For n > 1, a(n) gives the (one-based) index of the column where n is located in array A284311, or respectively, index of the row where n is in A284457. A008479 gives the other index.
%H A285329 Antti Karttunen, <a href="/A285329/b285329.txt">Table of n, a(n) for n = 1..10000</a>
%F A285329 a(n) = A013928(A007947(n)).
%F A285329 Other identities. For all n >= 0:
%F A285329 If A008683(n) <> 0 [when n is squarefree, A005117], a(n) = A013928(n), otherwise a(n) = a(A285328(n)).
%F A285329 a(A019565(n)) = A064273(n).
%o A285329 (Scheme, with memoization-macro definec)
%o A285329 (definec (A285329 n) (if (not (zero? (A008683 n))) (A013928 n) (A285329 (A285328 n))))
%o A285329 (define (A285329 n) (A013928 (A007947 n)))
%o A285329 (Python)
%o A285329 from operator import mul
%o A285329 from sympy import primefactors
%o A285329 from sympy.ntheory.factor_ import core
%o A285329 from functools import reduce
%o A285329 def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
%o A285329 def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)
%o A285329 print([a013928(a007947(n)) for n in range(1, 101)]) # _Indranil Ghosh_, Apr 18 2017
%o A285329 (Python)
%o A285329 from math import prod, isqrt
%o A285329 from sympy import primefactors, mobius
%o A285329 def A285329(n):
%o A285329     m=prod(primefactors(n))-1
%o A285329     return sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # _Chai Wah Wu_, May 12 2024
%Y A285329 Cf. A008479 (the other index).
%Y A285329 Cf. A005117, A007947, A008683, A013928, A019565, A064273, A087207, A285328.
%Y A285329 Cf. array A284311 (A284457).
%K A285329 nonn
%O A285329 1,3
%A A285329 _Antti Karttunen_, Apr 17 2017