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%I #18 Aug 10 2022 22:35:28
%S 1,1,2,1,9,2,125,1,4,9,2401,2,161051,125,18,1,4826809,4,410338673,9,
%T 250,2401,16983563041,2,81,161051,8,125,1801152661463,18,
%U 420707233300201,1,4802,4826809,1125,4,25408476896404831,410338673,322102,9
%N 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.
%C 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).
%C 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
%D Joseph J. Rotman, The Theory of Groups: An Introduction, 2nd ed. Boston: Allyn and Bacon, Inc. 1973. Page 9, problem 1.26.
%H Andrew Howroyd, <a href="/A090884/b090884.txt">Table of n, a(n) for n = 1..500</a>
%H Sam Alexander, <a href="http://tinyurl.com/yzjw">Post to sci.math</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Transport_of_structure">Transport of structure</a>
%e 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
%o (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
%Y Cf. A001222, A048675, A054841, A090880, A090881, A090882, A090883.
%Y Polynomial multiplication using the same isomorphism: A297845.
%K easy,nonn,mult
%O 1,3
%A _Sam Alexander_, Dec 12 2003
%E More terms from _Ray Chandler_, Dec 20 2003
%E New name from _Peter Munn_, Aug 10 2022 using existing formula (_Andrew Howroyd_, Jul 31 2018) and introductory comment.
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