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a(n) = A001222(A048675(n)).
11

%I #38 Jan 11 2022 10:48:25

%S 0,1,1,2,1,3,1,2,1,4,2,5,2,2,2,6,1,7,2,2,1,8,1,3,2,2,2,9,1,10,1,3,2,3,

%T 2,11,2,2,1,12,1,13,3,3,1,14,2,4,2,3,2,15,1,3,1,3,4,16,3,17,3,3,2,4,1,

%U 18,3,3,1,19,1,20,2,2,3,4,2,21,3,3,2,22,3,3,2,2,1,23,2,4,3,5,3,4,1,24,1,3,2,25,1,26,2,2

%N a(n) = A001222(A048675(n)).

%C For n >= 3, a(n) = index of the row where n is located in array A277898.

%H Antti Karttunen (terms 2..3465) and Hans Havermann, <a href="/A277892/b277892.txt">Table of n, a(n) for n = 2..10000</a>

%F a(A019565(n)) = a(A260443(n)) = A001222(n).

%F For all n >= 2, a(A065091(n)) = n.

%t A048675[n_] := If[n == 1, 0, Total[#[[2]]*2^(PrimePi[#[[1]]] - 1)& /@ FactorInteger[n]]];

%t a[n_] := PrimeOmega[A048675[n]];

%t Table[a[n], {n, 2, 105}] (* _Jean-François Alcover_, Jan 11 2022 *)

%o (Scheme) (define (A277892 n) (if (= 1 n) 0 (A001222 (A048675 n))))

%o (PARI)

%o A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2;

%o A277892(n) = if(1==n,0,bigomega(A048675(n)));

%o for(n=1, 3465, write("b277892.txt", n, " ", A277892(n)));

%o (Python)

%o from sympy import factorint, primepi, primefactors

%o def a001222(n): return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1

%o def a048675(n):

%o if n==1: return 0

%o f=factorint(n)

%o return sum(f[i]*2**(primepi(i) - 1) for i in f)

%o def a(n): return a001222(a048675(n))

%o print([a(n) for n in range(2, 101)]) # _Indranil Ghosh_, Jun 19 2017

%Y Left inverse of A065091.

%Y Cf. A001222, A048675.

%Y Cf. A277319 (positions of ones).

%Y Cf. A000040 (positions of records), A277900.

%Y Cf. A277895 (ordinal transform from a(3) onward).

%Y Cf. A277893, A277894, A277898.

%K nonn

%O 2,4

%A _Antti Karttunen_, Nov 08 2016