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A277322 a(n) = number of irreducible polynomial factors (counted with multiplicity) in the polynomial (with nonnegative integral coefficients) constructed from the prime factorization of n. 8
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..30033

FORMULA

a(2^n) = 0. [By an explicit convention.]

a(A000040(n)) = n-1.

a(A007188(n)) = n.

a(A260443(n)) = A277013(n).

EXAMPLE

For n = 7 = prime(4), the corresponding polynomial is x^3, which factorizes as (x)(x)(x), thus a(7) = 3.

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.

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.

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.

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.

PROG

(PARI)

allocatemem(2^29);

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)};

pfps(n) = if(1==n, 0, if(!(n%2), 1 + pfps(n/2), 'x*pfps(A064989(n))));

A277322 = n -> if(!bitand(n, (n-1)), 0, vecsum(factor(pfps(n))[, 2]));

for(n=1, 121121, write("b277322.txt", n, " ", A277322(n)));

(PARI)

pfps(n)=my(f=factor(n)); sum(i=1, #f~, f[i, 2] * 'x^(primepi(f[i, 1])-1))

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

CROSSREFS

Cf. A206442 (gives the number of irreducible polynomial factors without multiplicity), A206284 (positions of 1's, i.e., irreducible polynomials).

Cf. A000040, A007188, A064989, A260443, A277013.

Sequence in context: A127139 A166139 A071431 * A182740 A228786 A140699

Adjacent sequences:  A277319 A277320 A277321 * A277323 A277324 A277325

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 09 2016

STATUS

approved

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Last modified January 21 11:00 EST 2019. Contains 319351 sequences. (Running on oeis4.)