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%I #10 Aug 24 2019 11:56:36
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,
%T 0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,-1,1,0,1,0,0,0,0,0,-1,0,0,1,0,
%U 0,2,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,1,1,0,1,0,0,1,1,0,3,0,0,0,0,0,-1,0,0,1,0,0,1
%N Difference between the number of prime divisors in a nonstandard factorization process based on the sieve of Eratosthenes vs. their number in the ordinary factorization of n (when counted with multiplicity): a(n) = A253557(n) - A001222(n).
%H Antti Karttunen, <a href="/A326075/b326075.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A253557(n) - A001222(n) = A001222(A250246(n)) - A001222(n).
%F a(p) = 0 for all primes p.
%e A001222(21) = 2 because A032742(21) = 7, and A032742(7) = 1, while A253557(21) = 3 because A302042(21) = 9, A302042(9) = 3, and A302042(3) = 1. Thus a(21) = 3-2 = 1.
%e A001222(27) = 3 because A032742(27) = 9, A032742(9) = 3 and A032742(3) = 1, while A253557(27) = 2 because A302042(27) = 7 and A302042(7) = 1. Thus a(27) = 2-3 = -1.
%o (PARI)
%o up_to = 65537;
%o ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
%o v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A078898(n) = v078898[n];
%o A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
%o A253557(n) = if(1==n, 0,1+A253557(A302042(n)));
%o A326075(n) = (A253557(n)-bigomega(n));
%Y Cf. A001222, A032742, A250246, A253557, A302042, A323888, A326139, A326189, A326190, A326191.
%K sign
%O 1,69
%A _Antti Karttunen_, Aug 23 2019
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