A300249 Filter sequence combining A003415(n) and A046523(n), the arithmetic derivative of n and the prime signature of n.

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

Offset: 1

Author

Antti Karttunen, Mar 04 2018

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A046523.

Cf. also A300226, A300229, A300248.

Differs from A300235 for the first time at n=105, where a(105)=56, while A300235(105)=75.

Programs

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
Aux300249(n) = ((1/2)*(2 + ((A003415(n)+A046523(n))^2) - A003415(n) - 3*A046523(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300248(n))),"b300249.txt");