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A090884
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a(n) is the derivative of n via transport of structure from polynomials. Completely multiplicative with a(2) = 1, a(prime(i+1)) = prime(i)^i for i > 0.
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8
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1, 1, 2, 1, 9, 2, 125, 1, 4, 9, 2401, 2, 161051, 125, 18, 1, 4826809, 4, 410338673, 9, 250, 2401, 16983563041, 2, 81, 161051, 8, 125, 1801152661463, 18, 420707233300201, 1, 4802, 4826809, 1125, 4, 25408476896404831, 410338673, 322102, 9
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OFFSET
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1,3
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COMMENTS
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Previous name: There exists an isomorphism from the positive rationals under multiplication to Z[x] under addition, defined by f(q) = e1 + (e2)x + (e3)(x^2) +...+ (ek)(x^(k-1)) + ... (where e_i is the exponent of the i-th prime in q's prime factorization) The a(n) above are calculated by a(n) = f^(-1)[d/dx f(n)] (In other words: differentiate n's image in Z[x] and return to Q).
With primes noted p_0 = 2, p_1 = 3, etc., let f be the function that maps n = Product_{i=0..d} p_i^e_i to P = Sum_{i=0..d} e_i*X^i; and let g be the inverse function of f. a(n) is by definition g(P') = g((f(n))'). - Luc Rousseau, Aug 06 2018
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REFERENCES
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Joseph J. Rotman, The Theory of Groups: An Introduction, 2nd ed. Boston: Allyn and Bacon, Inc. 1973. Page 9, problem 1.26.
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LINKS
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EXAMPLE
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504 = 2^3 * 3^2 * 7 is mapped to polynomial 3+2X+X^3, whose derivative is 2+3X^2, which is mapped to 2^2 * 5^3 = 500. Then, a(504) = 500. - Luc Rousseau, Aug 06 2018
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PROG
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(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); if(p==2, 1, precprime(p-1)^(e*primepi(p-1))))} \\ Andrew Howroyd, Jul 31 2018
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CROSSREFS
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Polynomial multiplication using the same isomorphism: A297845.
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KEYWORD
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easy,nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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