A090883 Write n as Product_{i=1..k} prime(i)^e_i, where prime(i) is the i-th prime number and e_i is a nonnegative integer. a(n) = Sum_{i=1..k} e_i*n^(i-1).

0, 1, 3, 2, 25, 7, 343, 3, 18, 101, 14641, 14, 371293, 2745, 240, 4, 24137569, 37, 893871739, 402, 9282, 234257, 78310985281, 27, 1250, 11881377, 81, 21954, 14507145975869, 931, 819628286980801, 5, 1185954, 1544804417, 44100, 74

Offset: 1

Author

Sam Alexander, Dec 12 2003

Keywords

• easy
• nonn

Comments

In the definition, replace "e_i*n^(i-1)" with "e_i*x^(i-1)" for all i to define a function P:N+ -> N[x]. If we extend this definition to positive rationals by allowing negative e_i, P(.) becomes an isomorphism between positive rationals under multiplication and polynomials over Z under addition. We can use P to generalize A001222, A048675 and A054841: evaluate each term of the sequence of polynomials P(1), P(2), ... at x=1, x=2 and x=10, respectively. [Edited and corrected by Peter Munn, Aug 12 2022]

References

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

Links

• Sam Alexander, Post to sci.math. [Broken link]

Crossrefs

The main diagonal of A104244 (A104245).

Cf. A001222, A048675, A054841, A090880, A090881, A090882, A090884.

Programs

PARI

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*n^(primepi(f[k,1])-1)); \\ Michel Marcus, Nov 01 2016

Extensions

Name edited by Peter Munn, Aug 12 2022