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^(k-1)) + ...

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

Offset: 1

Author

Sam Alexander, Dec 12 2003

Keywords

• easy
• nonn

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.

Crossrefs

Row 3 of A104244.

Cf. A001222, A006190, A048675, A054841, A090881, A090882, A090883, A090884, A178590, A206296, A260443.

Programs

Scheme

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

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).

Extensions

More terms from Ray Chandler, Dec 20 2003