A090884 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.

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

Offset: 1

Author

Sam Alexander, Dec 12 2003

Keywords

• easy
• nonn
• mult

Comments

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

Examples

			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
		

References

• Joseph J. Rotman, The Theory of Groups: An Introduction, 2nd ed. Boston: Allyn and Bacon, Inc. 1973. Page 9, problem 1.26.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Sam Alexander, Post to sci.math.
• Wikipedia, Transport of structure

Crossrefs

Cf. A001222, A048675, A054841, A090880, A090881, A090882, A090883.

Polynomial multiplication using the same isomorphism: A297845.

Programs

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

Extensions

More terms from Ray Chandler, Dec 20 2003

New name from Peter Munn, Aug 10 2022 using existing formula (Andrew Howroyd, Jul 31 2018) and introductory comment.