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A300249
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Filter sequence combining A003415(n) and A046523(n), the arithmetic derivative of n and the prime signature of n.
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8
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1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67
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OFFSET
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1,2
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COMMENTS
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Restricted growth sequence transform of P(A003415(n), A046523(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.
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LINKS
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EXAMPLE
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a(51) = a(91) (= 33) because both are nonsquare semiprimes (3*17 and 7*13), and also their arithmetic derivatives are equal, as A003415(51) = A003415(91) = 20.
a(78) = a(105) (= 56) because both have the same prime signature (78 = 2*3*13 and 105 = 3*5*7), and also their arithmetic derivatives are equal, as A003415(78) = A003415(105) = 71.
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PROG
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(PARI)
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From A046523
write_to_bfile(1, rgs_transform(vector(65537, n, Aux300248(n))), "b300249.txt");
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CROSSREFS
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Differs from A300235 for the first time at n=105, where a(105)=56, while A300235(105)=75.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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