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%I #11 Apr 02 2018 21:20:13
%S 0,1,2,2,3,3,4,3,4,4,5,4,6,5,5,4,7,5,8,5,6,6,9,5,6,7,6,6,10,6,11,5,7,
%T 8,7,6,12,9,7,6,13,7,14,7,8,10,15,6,8,7,8,8,16,7,9,7,8,11,17,7,18,12,
%U 8,6,8,8,19,9,9,8,20,7,21,13,9,10,9,8,22,7,9,14,23,8,10,15,9,8,24,9,12,11,10,16,9,7,25,9,10,8,26,9,27,9,10
%N Analog of A056239 for nonstandard factorization based on the sieve of Eratosthenes (A083221).
%C Each n occurs A000041(n) times in total.
%H Antti Karttunen, <a href="/A302039/b302039.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F a(1) = 0; for n > 1, a(n) = A055396(n) + a(A302042(n)).
%F a(1) = 0; for n > 1, a(n) = (A055396(n)*A302045(n)) + a(A302044(n)).
%F a(n) = A056239(A250246(n)).
%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, by Hasler.
%o A055396(n) = if(1==n,0,primepi(A020639(n)));
%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 A302039(n) = if(1==n,0,A055396(n) + A302039(A302042(n)));
%Y Cf. A000041, A056239, A250246, A302042, A302044, A302045.
%Y Cf. also A253557, A302041, A302050, A302051, A302052, A302055 for other similar analogs.
%K nonn
%O 1,3
%A _Antti Karttunen_, Mar 31 2018
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