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a(n) = A013928(A007947(n)).
7

%I #24 May 12 2024 02:01:16

%S 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 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 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 a(n) = A013928(A007947(n)).

%C 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 Antti Karttunen, <a href="/A285329/b285329.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A013928(A007947(n)).

%F Other identities. For all n >= 0:

%F If A008683(n) <> 0 [when n is squarefree, A005117], a(n) = A013928(n), otherwise a(n) = a(A285328(n)).

%F a(A019565(n)) = A064273(n).

%o (Scheme, with memoization-macro definec)

%o (definec (A285329 n) (if (not (zero? (A008683 n))) (A013928 n) (A285329 (A285328 n))))

%o (define (A285329 n) (A013928 (A007947 n)))

%o (Python)

%o from operator import mul

%o from sympy import primefactors

%o from sympy.ntheory.factor_ import core

%o from functools import reduce

%o def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))

%o def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)

%o print([a013928(a007947(n)) for n in range(1, 101)]) # _Indranil Ghosh_, Apr 18 2017

%o (Python)

%o from math import prod, isqrt

%o from sympy import primefactors, mobius

%o def A285329(n):

%o m=prod(primefactors(n))-1

%o return sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # _Chai Wah Wu_, May 12 2024

%Y Cf. A008479 (the other index).

%Y Cf. A005117, A007947, A008683, A013928, A019565, A064273, A087207, A285328.

%Y Cf. array A284311 (A284457).

%K nonn

%O 1,3

%A _Antti Karttunen_, Apr 17 2017