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Write n as Product_{i=1..k} prime(i)^e_i, where prime(i) is the i-th prime number and e_i is a nonnegative integer. a(n) = Sum_{i=1..k} e_i*n^(i-1).
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%I #17 Aug 17 2022 22:28:41

%S 0,1,3,2,25,7,343,3,18,101,14641,14,371293,2745,240,4,24137569,37,

%T 893871739,402,9282,234257,78310985281,27,1250,11881377,81,21954,

%U 14507145975869,931,819628286980801,5,1185954,1544804417,44100,74

%N Write n as Product_{i=1..k} prime(i)^e_i, where prime(i) is the i-th prime number and e_i is a nonnegative integer. a(n) = Sum_{i=1..k} e_i*n^(i-1).

%C In the definition, replace "e_i*n^(i-1)" with "e_i*x^(i-1)" for all i to define a function P:N+ -> N[x]. If we extend this definition to positive rationals by allowing negative e_i, P(.) becomes an isomorphism between positive rationals under multiplication and polynomials over Z under addition. We can use P to generalize A001222, A048675 and A054841: evaluate each term of the sequence of polynomials P(1), P(2), ... at x=1, x=2 and x=10, respectively. [Edited and corrected by _Peter Munn_, Aug 12 2022]

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

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

%o (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

%Y The main diagonal of A104244 (A104245).

%Y Cf. A001222, A048675, A054841, A090880, A090881, A090882, A090884.

%K easy,nonn

%O 1,3

%A _Sam Alexander_, Dec 12 2003

%E Name edited by _Peter Munn_, Aug 12 2022