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A302051 An analog of A000005 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

%I A302051 #11 Apr 03 2018 15:10:53
%S A302051 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,6,4,
%T A302051 4,9,2,4,4,8,2,8,2,6,5,4,2,10,3,6,6,6,2,8,4,8,6,4,2,12,2,4,4,7,4,12,2,
%U A302051 6,8,8,2,12,2,4,4,6,4,8,2,10,6,4,2,12,6,4,8,8,2,10,4,6,6,4,4,12,2,6,4,9,2,12,2,8,9
%N A302051 An analog of A000005 for nonstandard factorization based on the sieve of Eratosthenes (A083221).
%C A302051 See A302042, A302044 and A302045 for a description of the factorization process.
%H A302051 Antti Karttunen, <a href="/A302051/b302051.txt">Table of n, a(n) for n = 1..65537</a>
%H A302051 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F A302051 a(1) = 1, for n > 1, a(n) = (A302045(n)+1) * a(A302044(n)).
%F A302051 a(n) = A000005(A250246(n)).
%F A302051 a(n) = A106737(A252754(n)).
%o A302051 (PARI)
%o A302051 up_to = 65537;
%o A302051 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 A302051 A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
%o A302051 v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A302051 A078898(n) = v078898[n];
%o A302051 A000265(n) = (n/2^valuation(n, 2));
%o A302051 A001511(n) = 1+valuation(n,2);
%o A302051 A302045(n) = A001511(A078898(n));
%o A302051 A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(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 A302051 A302051(n) = if(1==n,n,(A302045(n)+1)*A302051(A302044(n)));
%o A302051 (PARI)
%o A302051 \\ Or, using also some of the code from above:
%o A302051 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A302051 A055396(n) = if(1==n,0,primepi(A020639(n)));
%o A302051 A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
%o A302051 A302051(n) = numdiv(A250246(n));
%Y A302051 Cf. A000005, A083221, A302042, A302044, A302045, A302052 (reduced modulo 2), A302053 (gives the positions of odd numbers).
%Y A302051 Cf. also A253557, A302041, A302050, A302052, A302039, A302055 for other similar analogs.
%K A302051 nonn
%O A302051 1,2
%A A302051 _Antti Karttunen_, Apr 01 2018