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A296857
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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).
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3
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1, 2, 7, 16, 23, 126, 53, 512, 2401, 1150, 97, 9072, 151, 5194, 27209, 65536, 227, 388962, 311, 230000, 133931, 23474, 419, 2612736, 279841, 51038, 40353607, 2036048, 541, 12244050, 661, 33554432, 571039, 131206, 1668811, 252047376, 827, 224542, 1447033
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OFFSET
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1,2
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COMMENTS
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This sequence is the main diagonal of A248601.
See A248601 for additional comments.
For any n > 0, gcd(2 * n, a(2 * n)) = 2 * n.
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LINKS
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FORMULA
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For any n > 0 and k >= 0:
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EXAMPLE
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For n = 12:
- f(12) = (2, 1, 0, 0, ...),
- f(12) * f(12) = (4, 4, 0, 1, 0, 0, ...),
- a(12) = prime(1)^4 * prime(2)^4 * prime(4) = 2^4 * 3^4 * 7 = 9072.
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PROG
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(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])))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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