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A284010 a(n) = least natural number with the same prime signature polynomial p(n,x) has when it is factored over Z. Polynomial p(n,x) has only nonnegative integer coefficients that are encoded in the prime factorization of n.

%I A284010 #13 Mar 20 2017 23:22:23
%S A284010 0,0,2,0,4,2,8,0,2,2,16,2,32,6,6,0,64,2,128,2,6,2,256,2,4,6,2,2,512,2,
%T A284010 1024,0,30,6,12,2,2048,6,6,2,4096,2,8192,2,6,2,16384,2,8,2,30,2,32768,
%U A284010 2,12,2,30,30,65536,2,131072,6,6,0,60,2,262144,2,30,2,524288,2,1048576,6,6,2,24,6,2097152,2,2,6,4194304,6,12,6,6,2,8388608,4,24,2,210
%N A284010 a(n) = least natural number with the same prime signature polynomial p(n,x) has when it is factored over Z. Polynomial p(n,x) has only nonnegative integer coefficients that are encoded in the prime factorization of n.
%C A284010 Let p(n,x) be the completely additive polynomial-valued function such that p(prime(n),x) = x^(n-1) as defined by _Clark Kimberling_ in A206284. To compute a(n), factor p(n,x) over Z and collect the exponents of its irreducible polynomial factors using them as exponents of primes (in Z) as 2^e1 * 3^e2 * 5^e3 * ..., with e1 >= e2 >= e3 >= ...
%H A284010 Antti Karttunen, <a href="/A284010/b284010.txt">Table of n, a(n) for n = 1..8192</a>
%F A284010 a(2^n) = 0. [By an explicit convention.]
%F A284010 Other identities. For all n >= 1:
%F A284010 A284011(n) = a(A260443(n)).
%e A284010 For n = 7 = prime(4), the corresponding polynomial is x^3, which factorizes as (x)(x)(x), thus a(7) = 2^3 = 8.
%e A284010 For n = 14 = prime(4) * prime(1), the corresponding polynomial is x^3 + 1, which factorizes as (x + 1)(x^2 - x + 1), thus a(14) = 2^1 * 3^1 = 6.
%e A284010 For n = 90 = prime(3) * prime(2)^2 * prime(1), the corresponding polynomial is x^2 + 2x + 1, which factorizes as (x + 1)^2, thus a(90) = 2^2 = 4.
%o A284010 (PARI)
%o A284010 \\ After _Charles R Greathouse IV_'s code in A046523 and A277322:
%o A284010 pfps(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2] * 'x^(primepi(f[i, 1])-1)); };
%o A284010 A284010(n) = { if(!bitand(n, (n-1)), 0, my(p=0, f=vecsort(factor(pfps(n))[, 2], ,4)); prod(i=1, #f, (p=nextprime(p+1))^f[i])); }
%Y A284010 Cf. A046523, A206284 (positions of 2's), A206442, A277322, A284011, A284012.
%Y A284010 Cf. also A260443, A278233, A278243.
%K A284010 nonn
%O A284010 1,3
%A A284010 _Antti Karttunen_, Mar 20 2017