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A302045 a(1) = 0, for n > 1, a(n) = A001511(A078898(n)); Number of instances of the smallest prime factor A020639(n) in nonstandard factorization of n that is based on the sieve of Eratosthenes (A083221).

%I A302045 #20 Apr 05 2018 20:33:51
%S A302045 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,1,1,2,3,1,1,3,2,1,1,2,1,1,1,5,2,1,
%T A302045 1,2,1,1,1,3,1,1,1,2,4,1,1,4,2,1,1,2,1,1,3,3,2,1,1,2,1,1,1,6,1,1,1,2,
%U A302045 3,1,1,3,1,1,1,2,1,1,1,4,2,1,1,2,2,1,1,3,1,1,3,2,5,1,1,5,1,1,1,2,1,1,1,3,2
%N A302045 a(1) = 0, for n > 1, a(n) = A001511(A078898(n)); Number of instances of the smallest prime factor A020639(n) in nonstandard factorization of n that is based on the sieve of Eratosthenes (A083221).
%C A302045 Iterating the map n -> A302044(n) until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a sequence of distinct primes in ascending order, while applying this function (A302045) to those terms gives the corresponding "exponents" of those primes, that is, the count of consecutive occurrences of each prime when iterating the map n -> A302042(n), which gives the same primes with repetitions. Permutation pair A250245/A250246 maps between this non-standard prime factorization of n and the ordinary factorization of n. See also comments and examples in A302042.
%H A302045 Antti Karttunen, <a href="/A302045/b302045.txt">Table of n, a(n) for n = 1..65537</a>
%H A302045 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F A302045 a(1) = 0, for n > 1, a(n) = A001511(A078898(n)).
%F A302045 For n > 1, a(n) = A250245(A067029(A250246(n))).
%o A302045 (PARI)
%o A302045 A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
%o A302045 A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
%o A302045 A001511(n) = 1+valuation(n,2);
%o A302045 A302045(n) = if(1==n,0,A001511(A078898(n)));
%Y A302045 Cf. A001511, A067029, A078898, A250245, A250246, A268674, A302042, A302044.
%Y A302045 Cf. also A302035, A302050, A302051.
%K A302045 nonn
%O A302045 1,4
%A A302045 _Antti Karttunen_, Mar 31 2018