|
%I #27 Mar 24 2017 00:47:59
%S 0,0,1,0,2,1,3,0,1,1,4,1,5,2,2,0,6,1,7,1,2,1,8,1,2,2,1,1,9,1,10,0,3,2,
%T 3,1,11,2,2,1,12,1,13,1,2,1,14,1,3,1,3,1,15,1,3,1,3,3,16,1,17,2,2,0,4,
%U 1,18,1,3,1,19,1,20,2,2,1,4,2,21,1,1,2,22,2,3,2,2,1,23,2,4,1,4,2,4,1,24,1,2,1,25,1,26,1,2
%N a(n) = number of irreducible polynomial factors (counted with multiplicity) in the polynomial (with nonnegative integral coefficients) constructed from the prime factorization of n.
%C 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. Then this sequence is the number of irreducible factors in p(n,x), counted with multiplicity.
%H Antti Karttunen, <a href="/A277322/b277322.txt">Table of n, a(n) for n = 1..30033</a>
%F a(2^n) = 0. [By an explicit convention.]
%F a(A000040(n)) = n-1.
%F a(A007188(n)) = n.
%F a(A260443(n)) = A277013(n).
%e For n = 7 = prime(4), the corresponding polynomial is x^3, which factorizes as (x)(x)(x), thus a(7) = 3.
%e 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.
%e 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.
%e pfps(660) = pfps(2^2*3*5*11) = pfps(2^2) + pfps(3) + pfps(5) + pfps(11) = 2 + x + x^2 + x^4 which is irreducible, so a(660) = 1.
%e For n = 30030 = Product_{i=1..6} prime(i), the corresponding polynomial is x^5 + x^4 + x^3 + x^2 + x + 1, which factorizes as (x+1)(x^2 - x + 1)(x^2 + x + 1), thus a(30030) = 3.
%o (PARI)
%o allocatemem(2^29);
%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o pfps(n) = if(1==n, 0, if(!(n%2), 1 + pfps(n/2), 'x*pfps(A064989(n))));
%o A277322 = n -> if(!bitand(n,(n-1)), 0, vecsum(factor(pfps(n))[,2]));
%o for(n=1, 121121, write("b277322.txt", n, " ", A277322(n)));
%o (PARI)
%o pfps(n)=my(f=factor(n)); sum(i=1,#f~, f[i,2] * 'x^(primepi(f[i,1])-1))
%o A277322(n) = if(1==n, 0, vecsum(factor(pfps(n))[, 2])); \\ _Charles R Greathouse IV_, test for one added by _Antti Karttunen_, Oct 09 2016
%Y Cf. A206442 (gives the number of irreducible polynomial factors without multiplicity), A206284 (positions of 1's, i.e., irreducible polynomials).
%Y Cf. A000040, A007188, A064989, A260443, A277013.
%K nonn
%O 1,5
%A _Antti Karttunen_, Oct 09 2016
|
