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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)) + ... 9

%I #11 Apr 28 2022 07:59:10

%S 0,1,5,2,25,6,125,3,10,26,625,7,3125,126,30,4,15625,11,78125,27,130,

%T 626,390625,8,50,3126,15,127,1953125,31,9765625,5,630,15626,150,12,

%U 48828125,78126,3130,28,244140625,131,1220703125,627,35,390626,6103515625,9,250

%N 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)) + ...

%C 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.

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

%H Antti Karttunen, <a href="/A090882/b090882.txt">Table of n, a(n) for n = 1..1001</a>

%H Sam Alexander, <a href="http://tinyurl.com/yzjw">Post to sci.math</a>.

%o (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

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

%K easy,nonn

%O 1,3

%A _Sam Alexander_, Dec 12 2003

%E More terms from _Ray Chandler_, Dec 20 2003

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)