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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. 8
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, 1024, 0, 30, 6, 12, 2, 2048, 6, 6, 2, 4096, 2, 8192, 2, 6, 2, 16384, 2, 8, 2, 30, 2, 32768, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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. 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 >= ...

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

FORMULA

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

Other identities. For all n >= 1:

A284011(n) = a(A260443(n)).

EXAMPLE

For n = 7 = prime(4), the corresponding polynomial is x^3, which factorizes as (x)(x)(x), thus a(7) = 2^3 = 8.

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.

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.

PROG

(PARI)

\\ After Charles R Greathouse IV's code in A046523 and A277322:

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

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])); }

CROSSREFS

Cf. A046523, A206284 (positions of 2's), A206442, A277322, A284011, A284012.

Cf. also A260443, A278233, A278243.

Sequence in context: A163123 A194346 A328598 * A278082 A327442 A068773

Adjacent sequences:  A284007 A284008 A284009 * A284011 A284012 A284013

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 20 2017

STATUS

approved

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Last modified January 15 11:50 EST 2021. Contains 340187 sequences. (Running on oeis4.)