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

1,3

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

Table of n, a(n) for n=1..36.

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

Prog

(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

Crossrefs

The main diagonal of A104244 (A104245).

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

Sequence in context: A105525 A228772 A165714 * A100645 A132960 A009574

Adjacent sequences: A090880 A090881 A090882 * A090884 A090885 A090886

Keyword

easy,nonn

Author

Sam Alexander, Dec 12 2003

Extensions

Name edited by Peter Munn, Aug 12 2022

Status

approved