

A090883


Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)*n + (e3)*(n^2) + (e4)*(n^3) + ... + (ek)*(n^(k1)) + ...


8



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
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OFFSET

1,3


COMMENTS

Replace "(ek)*(n^(k1))" with "(ek)*(x^(k1))" for all k and extend the definition of a to positive rationals and a becomes an isomorphism between positive rationals under multiplication and polynomials over Z under addition. This remark generalizes A001222, A048675 and A054841: evaluate said polynomial at x=1, x=2 and x=10, respectively.


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


STATUS

approved



