%I
%S 0,1,2,0,4,3,8,0,0,0,16,0,32,0,6,0,64,5,128,0,0,0,256,0,0,0,0,0,512,7,
%T 1024,0,0,0,12,0,2048,0,0,0,4096,0,8192,0,0,0,16384,0,0,0,0,0,32768,0,
%U 0,0,0,0,65536,0,131072,0,0,0,0,0,262144,0,0,0,524288,0,1048576,0,10,0,24,0,2097152,0,0,0,4194304,0,0,0,0,0,8388608,9
%N Left inverse of A260443, giving 0 as a result when n is outside of the range of A260443.
%H Antti Karttunen, <a href="/A277333/b277333.txt">Table of n, a(n) for n = 1..5000</a>
%F If A260443(A048675(n)) = n, then a(n) = A048675(n), otherwise a(n) = 0.
%F Other identities. For all n >= 0:
%F a(A260443(n)) = n.
%F a(2n+1) = 2*a(A064989(2n+1)).
%F If a(2n) > 0 [by necessity an odd number in that case], then A005811((a(2n)1)/2) = A007949(2n). [See comment in A277324.]
%e a(1) = 0 because A260443(0) = 1. For n > 1, a(n) = 0 only if n does not occur in the range of A260443.
%e a(6) = 3 because A260443(3) = 6.
%o (PARI)
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From _Michel Marcus_
%o A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ _Michel Marcus_, Oct 10 2016
%o A260443(n) = if(n<2, n+1, if(!(n%2), A003961(A260443(n/2)), A260443((n1)/2) * A260443((n+1)/2)));
%o A277333(n) = { my(k=A048675(n)); if(A260443(k) == n, k, 0); } ;
%o for(n=1, 5000, write("b277333.txt", n, " ", A277333(n)));
%o (Scheme) (define (A277333 n) (let ((k (A048675 n))) (if (= (A260443 k) n) k 0)))
%Y Cf. A005811, A007949, A048675, A064989, A260443.
%Y Cf. A277316, A260442 (from 2 onward, the positions of nonzeros), A277317 (positions of primes).
%K nonn
%O 1,3
%A _Antti Karttunen_, Oct 10 2016
