%I #11 Apr 03 2018 15:11:01
%S 1,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
%T 0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1
%N Characteristic function for A302053; an analog of A010052 (char. fun of squares) for the nonstandard factorization based on the sieve of Eratosthenes (A083221).
%H Antti Karttunen, <a href="/A302052/b302052.txt">Table of n, a(n) for n = 0..65537</a>
%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F a(0) = 1, for n >= 1, a(n) = A302051(n) mod 2.
%F For n >= 1, a(n) = A010052(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.
%o v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A078898(n) = v078898[n];
%o A000265(n) = (n/2^valuation(n, 2));
%o A001511(n) = 1+valuation(n,2);
%o A302045(n) = A001511(A078898(n));
%o 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 A302052(n) = if(n<=1,1,if((A302045(n)%2),0,A302052(A302044(n))));
%o (PARI)
%o \\ Or, using also some of the code from above:
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A055396(n) = if(1==n,0,primepi(A020639(n)));
%o 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 A302052(n) = if(!n,1,issquare(A250246(n)));
%Y Cf. A010052, A250246, A302044, A302045, A302053 (positions of ones).
%Y Cf. also A253557, A302041, A302050, A302051, A302039, A302055 for other similar analogs.
%K nonn
%O 0
%A _Antti Karttunen_, Mar 31 2018