

A090880


Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)*3 + (e3)*9 + (e4)*27 + ... + (ek)*(3^(k1)) + ...


15



0, 1, 3, 2, 9, 4, 27, 3, 6, 10, 81, 5, 243, 28, 12, 4, 729, 7, 2187, 11, 30, 82, 6561, 6, 18, 244, 9, 29, 19683, 13, 59049, 5, 84, 730, 36, 8, 177147, 2188, 246, 12, 531441, 31, 1594323, 83, 15, 6562, 4782969, 7, 54, 19, 732, 245, 14348907, 10, 90, 30, 2190
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OFFSET

1,3


COMMENTS

Replace "3" with "x" 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.
For examples of such evaluations at x=3, see "Other identities" in the Formula section.  Antti Karttunen, Jul 31 2015


REFERENCES

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


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..512
Sam Alexander, Post to sci.math.


FORMULA

a(1) = 0; for n > 1, a(n) = 3^(A055396(n)1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]  Antti Karttunen, Jul 29 2015
Other identities. For all n >= 0:
a(A206296(n)) = A006190(n).
a(A260443(n)) = A178590(n).


PROG

(Scheme) (define (A090880 n) (if (= 1 n) ( n 1) (+ (A000244 ( (A055396 n) 1)) (A090880 (A032742 n))))) ;; Antti Karttunen, Jul 29 2015


CROSSREFS

Row 3 of A104244.
Cf. A001222, A006190, A048675, A054841, A090881, A090882, A090883, A090884, A178590, A206296, A260443.
Sequence in context: A266636 A182652 A251555 * A258439 A188926 A193980
Adjacent sequences: A090877 A090878 A090879 * A090881 A090882 A090883


KEYWORD

easy,nonn


AUTHOR

Sam Alexander, Dec 12 2003


EXTENSIONS

More terms from Ray Chandler, Dec 20 2003


STATUS

approved



