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A296857 For any number n > 0, let f(n) be the function that associates k to the prime(k)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the arithmetic functions with nonnegative integer values and a finite number of nonzero values; let g be the inverse of f; a(n) = g(f(n) * f(n)) (where i * j denotes the Dirichlet convolution of i and j).

%I A296857 #20 Dec 27 2017 01:43:54
%S A296857 1,2,7,16,23,126,53,512,2401,1150,97,9072,151,5194,27209,65536,227,
%T A296857 388962,311,230000,133931,23474,419,2612736,279841,51038,40353607,
%U A296857 2036048,541,12244050,661,33554432,571039,131206,1668811,252047376,827,224542,1447033
%N A296857 For any number n > 0, let f(n) be the function that associates k to the prime(k)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the arithmetic functions with nonnegative integer values and a finite number of nonzero values; let g be the inverse of f; a(n) = g(f(n) * f(n)) (where i * j denotes the Dirichlet convolution of i and j).
%C A296857 This sequence is the main diagonal of A248601.
%C A296857 See A248601 for additional comments.
%C A296857 For any n > 0, gcd(2 * n, a(2 * n)) = 2 * n.
%H A296857 Rémy Sigrist, <a href="/A296857/b296857.txt">Table of n, a(n) for n = 1..10000</a>
%H A296857 Rémy Sigrist, <a href="/A296857/a296857.png">Colored logarithmic scatterplot of the first 50000 terms</a> (where the color is function of A001222(n))
%H A296857 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>.
%F A296857 For any n > 0 and k >= 0:
%F A296857 - a(n) = A248601(n, n),
%F A296857 - A001221(a(n)) <= A001221(n)^2,
%F A296857 - A001222(a(n)) = A001222(n)^2,
%F A296857 - A055396(a(n)) = A055396(n)^2,
%F A296857 - A061395(a(n)) = A061395(n)^2,
%F A296857 - a(A000040(n)) = A011757(n),
%F A296857 - a(A000040(n)^k) = A011757(n)^(k^2).
%e A296857 For n = 12:
%e A296857 - f(12) = (2, 1, 0, 0, ...),
%e A296857 - f(12) * f(12) = (4, 4, 0, 1, 0, 0, ...),
%e A296857 - a(12) = prime(1)^4 * prime(2)^4 * prime(4) = 2^4 * 3^4 * 7 = 9072.
%o A296857 (PARI) a(n) = my (f=factor(n), p=apply(primepi, f[,1]~)); prod(i=1, #p, prod(j=1, #p, prime(p[i]*p[j])^(f[i,2]*f[j,2])))
%Y A296857 Cf. A000040, A001221, A001222, A055396, A061395, A248601.
%K A296857 nonn
%O A296857 1,2
%A A296857 _Rémy Sigrist_, Dec 21 2017