A090882 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then a(n) = e1 + (e2)*5 + (e3)*25 + (e4)*125 + ... + (ek)*(5^(k-1)) + ...

0, 1, 5, 2, 25, 6, 125, 3, 10, 26, 625, 7, 3125, 126, 30, 4, 15625, 11, 78125, 27, 130, 626, 390625, 8, 50, 3126, 15, 127, 1953125, 31, 9765625, 5, 630, 15626, 150, 12, 48828125, 78126, 3130, 28, 244140625, 131, 1220703125, 627, 35, 390626, 6103515625, 9, 250

Offset: 1

Author

Sam Alexander, Dec 12 2003

Keywords

• easy
• nonn

Comments

Replace "5" 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.

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..1001
• Sam Alexander, Post to sci.math.

Crossrefs

Cf. A001222, A048675, A054841, A090880, A090881, A090883, A090884, and also A195017, A248663, A276075, A276085.

Programs

PARI

A090882(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*5^(primepi(f[k, 1])-1)); }; \\ Antti Karttunen, Apr 28 2022

Extensions

More terms from Ray Chandler, Dec 20 2003